Conjugate transpose: Difference between revisions
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{{
[[Category:Matrices]]
Suppose that a [[matrix]] <big><math>M</math></big> contains [[Arithmetic/Complex|complex numbers]]. Then the [[wp:conjugate transpose|conjugate transpose]] of <math>M</math> is a matrix <math>M^H</math> containing the [[complex conjugate]]s of the [[matrix transposition]] of <math>M</math>.
::: <big><math>(M^H)_{ji} = \overline{M_{ij}}</math></big>
This means that row <big><math>j</math></big>, column <big><math>i</math></big> of the conjugate transpose equals the
<br>complex conjugate of row <big><math>i</math></big>, column <big><math>j</math></big> of the original matrix.
In the next list, <big><math>M</math></big> must also be a square matrix.
* A [[wp:Hermitian matrix|Hermitian matrix]] equals its own conjugate transpose: <math>M^H = M</math>.
* A [[wp:normal matrix|normal matrix]] is commutative in [[matrix multiplication|multiplication]] with its conjugate transpose: <math>M^HM = MM^H</math>.
* A [[wp:unitary matrix|unitary matrix]] has its [[inverse matrix|inverse]] equal to its conjugate transpose: <math>M^H = M^{-1}</math>. <br> This is true [[wikt:iff|iff]] <math>M^HM = I_n</math> and iff <math>MM^H = I_n</math>, where <math>I_n</math> is the identity matrix.
<br>
;Task:
Given some matrix of complex numbers, find its conjugate transpose.
Also determine if the matrix is a:
::* Hermitian matrix,
::* normal matrix, or
::* unitary matrix.
;See also:
* MathWorld entry: [http://mathworld.wolfram.com/ConjugateTranspose.html conjugate transpose]
* MathWorld entry: [http://mathworld.wolfram.com/HermitianMatrix.html Hermitian matrix]
* MathWorld entry: [http://mathworld.wolfram.com/NormalMatrix.html normal matrix]
* MathWorld entry: [http://mathworld.wolfram.com/UnitaryMatrix.html unitary matrix]
<br><br>
=={{header|11l}}==
{{trans|Nim}}
<syntaxhighlight lang="11l">-V eps = 1e-10
F to_str(m)
V r = ‘’
L(row) m
V i = L.index
r ‘’= I i == 0 {‘[’} E ‘ ’
L(val) row
V j = L.index
I j != 0
r ‘’= ‘ ’
r ‘’= ‘(#2.4, #2.4)’.format(val.real, val.imag)
r ‘’= I i == m.len - 1 {‘]’} E "\n"
R r
F conjugateTransposed(m)
V r = [[0i] * m.len] * m.len
L(i) 0 .< m.len
L(j) 0 .< m.len
r[j][i] = conjugate(m[i][j])
R r
F mmul(m1, m2)
V r = [[0i] * m1.len] * m1.len
L(i) 0 .< m1.len
L(j) 0 .< m1.len
L(k) 0 .< m1.len
r[i][j] += m1[i][k] * m2[k][j]
R r
F isHermitian(m)
L(i) 0 .< m.len
L(j) 0 .< m.len
I m[i][j] != conjugate(m[j][i])
R 0B
R 1B
F isEqual(m1, m2)
L(i) 0 .< m1.len
L(j) 0 .< m1.len
I m1[i][j] != m2[i][j]
R 0B
R 1B
F isNormal(m)
V h = conjugateTransposed(m)
R isEqual(mmul(m, h), mmul(h, m))
F isIdentity(m)
L(i) 0 .< m.len
L(j) 0 .< m.len
I i == j
I abs(m[i][j] - 1.0) > :eps
R 0B
E
I abs(m[i][j]) > :eps
R 0B
R 1B
F isUnitary(m)
V h = conjugateTransposed(m)
R isIdentity(mmul(m, h)) & isIdentity(mmul(h, m))
F test(m)
print(‘Matrix’)
print(‘------’)
print(to_str(m))
print(‘’)
print(‘Conjugate transposed’)
print(‘--------------------’)
print(to_str(conjugateTransposed(m)))
print(‘’)
print(‘Hermitian: ’(I isHermitian(m) {‘true’} E ‘false’))
print(‘Normal: ’(I isNormal(m) {‘true’} E ‘false’))
print(‘Unitary: ’(I isUnitary(m) {‘true’} E ‘false’))
V M2 = [[3.0 + 0.0i, 2.0 + 1.0i],
[2.0 - 1.0i, 1.0 + 0.0i]]
V M3 = [[1.0 + 0.0i, 1.0 + 0.0i, 0.0 + 0.0i],
[0.0 + 0.0i, 1.0 + 0.0i, 1.0 + 0.0i],
[1.0 + 0.0i, 0.0 + 0.0i, 1.0 + 0.0i]]
V SR2 = 1 / sqrt(2.0)
V SR2i = SR2 * 1i
V M4 = [[SR2 + 0.0i, SR2 + 0.0i, 0.0 + 0.0i],
[0.0 + SR2i, 0.0 - SR2i, 0.0 + 0.0i],
[0.0 + 0.0i, 0.0 + 0.0i, 0.0 + 1.0i]]
test(M2)
print("\n")
test(M3)
print("\n")
test(M4)</syntaxhighlight>
{{out}}
<pre>
Matrix
------
[( 3.0000, 0.0000) ( 2.0000, 1.0000)
( 2.0000, -1.0000) ( 1.0000, 0.0000)]
Conjugate transposed
--------------------
[( 3.0000, 0.0000) ( 2.0000, 1.0000)
( 2.0000, -1.0000) ( 1.0000, 0.0000)]
Hermitian: true
Normal: true
Unitary: false
Matrix
------
[( 1.0000, 0.0000) ( 1.0000, 0.0000) ( 0.0000, 0.0000)
( 0.0000, 0.0000) ( 1.0000, 0.0000) ( 1.0000, 0.0000)
( 1.0000, 0.0000) ( 0.0000, 0.0000) ( 1.0000, 0.0000)]
Conjugate transposed
--------------------
[( 1.0000, 0.0000) ( 0.0000, 0.0000) ( 1.0000, 0.0000)
( 1.0000, 0.0000) ( 1.0000, 0.0000) ( 0.0000, 0.0000)
( 0.0000, 0.0000) ( 1.0000, 0.0000) ( 1.0000, 0.0000)]
Hermitian: false
Normal: true
Unitary: false
Matrix
------
[( 0.7071, 0.0000) ( 0.7071, 0.0000) ( 0.0000, 0.0000)
( 0.0000, 0.7071) ( 0.0000, -0.7071) ( 0.0000, 0.0000)
( 0.0000, 0.0000) ( 0.0000, 0.0000) ( 0.0000, 1.0000)]
Conjugate transposed
--------------------
[( 0.7071, 0.0000) ( 0.0000, -0.7071) ( 0.0000, 0.0000)
( 0.7071, 0.0000) ( 0.0000, 0.7071) ( 0.0000, 0.0000)
( 0.0000, 0.0000) ( 0.0000, 0.0000) ( 0.0000, -1.0000)]
Hermitian: false
Normal: true
Unitary: true
</pre>
=={{header|Ada}}==
<
with Ada.Complex_Text_IO; use Ada.Complex_Text_IO;
with Ada.Numerics.Complex_Types; use Ada.Numerics.Complex_Types;
Line 63 ⟶ 231:
Put_Line("nmat:"); Examine(nmat); New_Line;
Put_Line("umat:"); Examine(umat);
end ConTrans;</
{{out}}
<pre>hmat:
Line 99 ⟶ 267:
Unitary?: TRUE</pre>
=={{header|
Uses the same test cases as the Ada sample.
<syntaxhighlight lang="algol68">BEGIN # find and classify the complex conjugate transpose of a complex matrix #
# returns the conjugate transpose of m #
OP CONJUGATETRANSPOSE = ( [,]COMPL m )[,]COMPL:
BEGIN
[ 2 LWB m : 2 UPB m, 1 LWB m : 1 UPB m ]COMPL result;
FOR i FROM 1 LWB m TO 1 UPB m DO
FOR j FROM 2 LWB m TO 2 UPB m DO
result[ j, i ] := CONJ m[ i, j ]
OD
OD;
result
END # CONJUGATETRANSPOSE # ;
# returns TRUE if m is an identity matrix, FALSE otherwise #
OP ISIDENTITY = ( [,]COMPL m )BOOL:
IF 1 LWB m /= 2 LWB m OR 1 UPB m /= 2 UPB m THEN
# non-square matrix #
FALSE
ELSE
# the matrix is square #
# returns TRUE IF v - e is nearly 0, FALSE Otherwise #
PROC nearly equal = ( COMPL v, REAL e )BOOL: ABS re OF v - e < 1e-14 AND ABS im OF v < 1e-14;
BOOL result := TRUE;
FOR i FROM 1 LWB m TO 1 UPB m WHILE result DO
IF result := nearly equal( m[ i, i ], 1 ) THEN
# the diagonal element is 1 - test the non-diagonals #
FOR j FROM 1 LWB m TO 1 UPB m WHILE result DO
IF i /= j THEN result := nearly equal( m[ i, j ], 0 ) FI
OD
FI
OD;
result
FI # ISIDENTITY # ;
# returns m multiplied by n #
PRIO X = 7;
OP X = ( [,]COMPL m, n )[,]COMPL:
BEGIN
[ 1 : 1 UPB m, 1 : 2 UPB n ]COMPL r;
FOR i FROM 1 LWB m TO 1 UPB m DO
FOR j FROM 2 LWB n TO 2 UPB n DO
r[ i, j ] := 0;
FOR k TO 2 UPB n DO
r[ i, j ] +:= m[ i, k ] * n[ k, j ]
OD
OD
OD;
r
END # X # ;
# prints the complex matris m #
PROC show matrix = ( [,]COMPL m )VOID:
FOR i FROM 1 LWB m TO 1 UPB m DO
print( ( " " ) );
FOR j FROM 2 LWB m TO 2 UPB m DO
print( ( "( ", fixed( re OF m[ i, j ], -8, 4 )
, ", ", fixed( im OF m[ i, j ], -8, 4 )
, "i )"
)
)
OD;
print( ( newline ) )
OD # show matrix # ;
# display the matrix m, its conjugate transpose and whether it is Hermitian, Normal and Unitary #
PROC show = ( [,]COMPL m )VOID:
BEGIN
[,]COMPL c = CONJUGATETRANSPOSE m;
[,]COMPL cm = c X m;
[,]COMPL mc = m X c;
print( ( "Matrix:", newline ) );
show matrix( m );
print( ( "Conjugate Transpose:", newline ) );
show matrix( c );
BOOL is normal = cm = mc;
BOOL is unitary = IF NOT is normal THEN FALSE
ELSE ISIDENTITY mc
FI;
print( ( IF c = m THEN "" ELSE "not " FI, "Hermitian; "
, IF is normal THEN "" ELSE "not " FI, "Normal; "
, IF is unitary THEN "" ELSE "not " FI, "Unitary"
, newline
)
);
print( ( newline ) )
END # show # ;
# test some matrices for Hermitian, Normal and Unitary #
show( ( ( ( 3.0000 I 0.0000 ), ( 2.0000 I 1.0000 ) )
, ( ( 2.0000 I -1.0000 ), ( 1.0000 I 0.0000 ) )
)
);
show( ( ( ( 1.0000 I 0.0000 ), ( 1.0000 I 0.0000 ), ( 0.0000 I 0.0000 ) )
, ( ( 0.0000 I 0.0000 ), ( 1.0000 I 0.0000 ), ( 1.0000 I 0.0000 ) )
, ( ( 1.0000 I 0.0000 ), ( 0.0000 I 0.0000 ), ( 1.0000 I 0.0000 ) )
)
);
REAL rh = sqrt( 0.5 );
show( ( ( ( rh I 0.0000 ), ( rh I 0.0000 ), ( 0.0000 I 0.0000 ) )
, ( ( 0.0000 I rh ), ( 0.0000 I - rh ), ( 0.0000 I 0.0000 ) )
, ( ( 0.0000 I 0.0000 ), ( 0.0000 I 0.0000 ), ( 0.0000 I 1.0000 ) )
)
)
END</syntaxhighlight>
{{out}}
<pre>
Matrix:
( 3.0000, 0.0000i )( 2.0000, 1.0000i )
( 2.0000, -1.0000i )( 1.0000, 0.0000i )
Conjugate Transpose:
( 3.0000, 0.0000i )( 2.0000, 1.0000i )
( 2.0000, -1.0000i )( 1.0000, 0.0000i )
Hermitian; Normal; not Unitary
Matrix:
( 1.0000, 0.0000i )( 1.0000, 0.0000i )( 0.0000, 0.0000i )
( 0.0000, 0.0000i )( 1.0000, 0.0000i )( 1.0000, 0.0000i )
( 1.0000, 0.0000i )( 0.0000, 0.0000i )( 1.0000, 0.0000i )
Conjugate Transpose:
( 1.0000, 0.0000i )( 0.0000, 0.0000i )( 1.0000, 0.0000i )
( 1.0000, 0.0000i )( 1.0000, 0.0000i )( 0.0000, 0.0000i )
( 0.0000, 0.0000i )( 1.0000, 0.0000i )( 1.0000, 0.0000i )
not Hermitian; Normal; not Unitary
Matrix:
( 0.7071, 0.0000i )( 0.7071, 0.0000i )( 0.0000, 0.0000i )
( 0.0000, 0.7071i )( 0.0000, -0.7071i )( 0.0000, 0.0000i )
( 0.0000, 0.0000i )( 0.0000, 0.0000i )( 0.0000, 1.0000i )
Conjugate Transpose:
( 0.7071, 0.0000i )( 0.0000, -0.7071i )( 0.0000, 0.0000i )
( 0.7071, 0.0000i )( 0.0000, 0.7071i )( 0.0000, 0.0000i )
( 0.0000, 0.0000i )( 0.0000, 0.0000i )( 0.0000, -1.0000i )
not Hermitian; Normal; Unitary</pre>
=={{header|C}}==
<syntaxhighlight lang="c">/* Uses C99 specified complex.h, complex datatype has to be defined and operation provided if used on non-C99 compilers */
#include<stdlib.h>
Line 125 ⟶ 420:
b.cols = a.rows;
b.z =
for (i = 0; i < b.rows; i++)
{
b.z[i] =
for (j = 0; j < b.cols; j++)
{
Line 174 ⟶ 469:
c.cols = b.cols;
c.z =
for (i = 0; i < c.rows; i++)
{
c.z[i] =
c.z[i][j] = 0 + 0 * I;
for (j = 0; j < b.cols; j++)
Line 247 ⟶ 542:
scanf ("%d%d", &a.rows, &a.cols);
a.z =
printf ("Randomly Generated Complex Matrix A is : ");
for (i = 0; i < a.rows; i++)
{
printf ("\n");
a.z[i] =
for (j = 0; j < a.cols; j++)
{
Line 266 ⟶ 561:
{
printf ("\n");
aT.z[i] =
for (j = 0; j < aT.cols; j++)
{
Line 284 ⟶ 579:
return 0;
}</syntaxhighlight>
{{out}}
<pre>
Line 305 ⟶ 599:
Complex Matrix A is not normal</pre>
=={{header|C++}}==
<syntaxhighlight lang="cpp">#include <cassert>
#include <cmath>
#include <complex>
#include <iomanip>
#include <iostream>
#include <sstream>
#include <vector>
template <typename scalar_type> class complex_matrix {
public:
using element_type = std::complex<scalar_type>;
complex_matrix(size_t rows, size_t columns)
: rows_(rows), columns_(columns), elements_(rows * columns) {}
complex_matrix(size_t rows, size_t columns, element_type value)
: rows_(rows), columns_(columns), elements_(rows * columns, value) {}
complex_matrix(size_t rows, size_t columns,
const std::initializer_list<std::initializer_list<element_type>>& values)
: rows_(rows), columns_(columns), elements_(rows * columns) {
assert(values.size() <= rows_);
size_t i = 0;
for (const auto& row : values) {
assert(row.size() <= columns_);
std::copy(begin(row), end(row), &elements_[i]);
i += columns_;
}
}
size_t rows() const { return rows_; }
size_t columns() const { return columns_; }
const element_type& operator()(size_t row, size_t column) const {
assert(row < rows_);
assert(column < columns_);
return elements_[row * columns_ + column];
}
element_type& operator()(size_t row, size_t column) {
assert(row < rows_);
assert(column < columns_);
return elements_[row * columns_ + column];
}
friend bool operator==(const complex_matrix& a, const complex_matrix& b) {
return a.rows_ == b.rows_ && a.columns_ == b.columns_ &&
a.elements_ == b.elements_;
}
private:
size_t rows_;
size_t columns_;
std::vector<element_type> elements_;
};
template <typename scalar_type>
complex_matrix<scalar_type> product(const complex_matrix<scalar_type>& a,
const complex_matrix<scalar_type>& b) {
assert(a.columns() == b.rows());
size_t arows = a.rows();
size_t bcolumns = b.columns();
size_t n = a.columns();
complex_matrix<scalar_type> c(arows, bcolumns);
for (size_t i = 0; i < arows; ++i) {
for (size_t j = 0; j < n; ++j) {
for (size_t k = 0; k < bcolumns; ++k)
c(i, k) += a(i, j) * b(j, k);
}
}
return c;
}
template <typename scalar_type>
complex_matrix<scalar_type>
conjugate_transpose(const complex_matrix<scalar_type>& a) {
size_t rows = a.rows(), columns = a.columns();
complex_matrix<scalar_type> b(columns, rows);
for (size_t i = 0; i < columns; i++) {
for (size_t j = 0; j < rows; j++) {
b(i, j) = std::conj(a(j, i));
}
}
return b;
}
template <typename scalar_type>
std::string to_string(const std::complex<scalar_type>& c) {
std::ostringstream out;
const int precision = 6;
out << std::fixed << std::setprecision(precision);
out << std::setw(precision + 3) << c.real();
if (c.imag() > 0)
out << " + " << std::setw(precision + 2) << c.imag() << 'i';
else if (c.imag() == 0)
out << " + " << std::setw(precision + 2) << 0.0 << 'i';
else
out << " - " << std::setw(precision + 2) << -c.imag() << 'i';
return out.str();
}
template <typename scalar_type>
void print(std::ostream& out, const complex_matrix<scalar_type>& a) {
size_t rows = a.rows(), columns = a.columns();
for (size_t row = 0; row < rows; ++row) {
for (size_t column = 0; column < columns; ++column) {
if (column > 0)
out << ' ';
out << to_string(a(row, column));
}
out << '\n';
}
}
template <typename scalar_type>
bool is_hermitian_matrix(const complex_matrix<scalar_type>& matrix) {
if (matrix.rows() != matrix.columns())
return false;
return matrix == conjugate_transpose(matrix);
}
template <typename scalar_type>
bool is_normal_matrix(const complex_matrix<scalar_type>& matrix) {
if (matrix.rows() != matrix.columns())
return false;
auto c = conjugate_transpose(matrix);
return product(c, matrix) == product(matrix, c);
}
bool is_equal(const std::complex<double>& a, double b) {
constexpr double e = 1e-15;
return std::abs(a.imag()) < e && std::abs(a.real() - b) < e;
}
template <typename scalar_type>
bool is_identity_matrix(const complex_matrix<scalar_type>& matrix) {
if (matrix.rows() != matrix.columns())
return false;
size_t rows = matrix.rows();
for (size_t i = 0; i < rows; ++i) {
for (size_t j = 0; j < rows; ++j) {
if (!is_equal(matrix(i, j), scalar_type(i == j ? 1 : 0)))
return false;
}
}
return true;
}
template <typename scalar_type>
bool is_unitary_matrix(const complex_matrix<scalar_type>& matrix) {
if (matrix.rows() != matrix.columns())
return false;
auto c = conjugate_transpose(matrix);
auto p = product(c, matrix);
return is_identity_matrix(p) && p == product(matrix, c);
}
template <typename scalar_type>
void test(const complex_matrix<scalar_type>& matrix) {
std::cout << "Matrix:\n";
print(std::cout, matrix);
std::cout << "Conjugate transpose:\n";
print(std::cout, conjugate_transpose(matrix));
std::cout << std::boolalpha;
std::cout << "Hermitian: " << is_hermitian_matrix(matrix) << '\n';
std::cout << "Normal: " << is_normal_matrix(matrix) << '\n';
std::cout << "Unitary: " << is_unitary_matrix(matrix) << '\n';
}
int main() {
using matrix = complex_matrix<double>;
matrix matrix1(3, 3, {{{2, 0}, {2, 1}, {4, 0}},
{{2, -1}, {3, 0}, {0, 1}},
{{4, 0}, {0, -1}, {1, 0}}});
double n = std::sqrt(0.5);
matrix matrix2(3, 3, {{{n, 0}, {n, 0}, {0, 0}},
{{0, -n}, {0, n}, {0, 0}},
{{0, 0}, {0, 0}, {0, 1}}});
matrix matrix3(3, 3, {{{2, 2}, {3, 1}, {-3, 5}},
{{2, -1}, {4, 1}, {0, 0}},
{{7, -5}, {1, -4}, {1, 0}}});
test(matrix1);
std::cout << '\n';
test(matrix2);
std::cout << '\n';
test(matrix3);
return 0;
}</syntaxhighlight>
{{out}}
<pre>
Matrix:
2.000000 + 0.000000i 2.000000 + 1.000000i 4.000000 + 0.000000i
2.000000 - 1.000000i 3.000000 + 0.000000i 0.000000 + 1.000000i
4.000000 + 0.000000i 0.000000 - 1.000000i 1.000000 + 0.000000i
Conjugate transpose:
2.000000 + 0.000000i 2.000000 + 1.000000i 4.000000 + 0.000000i
2.000000 - 1.000000i 3.000000 + 0.000000i 0.000000 + 1.000000i
4.000000 + 0.000000i 0.000000 - 1.000000i 1.000000 + 0.000000i
Hermitian: true
Normal: true
Unitary: false
Matrix:
0.707107 + 0.000000i 0.707107 + 0.000000i 0.000000 + 0.000000i
0.000000 - 0.707107i 0.000000 + 0.707107i 0.000000 + 0.000000i
0.000000 + 0.000000i 0.000000 + 0.000000i 0.000000 + 1.000000i
Conjugate transpose:
0.707107 + 0.000000i 0.000000 + 0.707107i 0.000000 + 0.000000i
0.707107 + 0.000000i 0.000000 - 0.707107i 0.000000 + 0.000000i
0.000000 + 0.000000i 0.000000 + 0.000000i 0.000000 - 1.000000i
Hermitian: false
Normal: true
Unitary: true
Matrix:
2.000000 + 2.000000i 3.000000 + 1.000000i -3.000000 + 5.000000i
2.000000 - 1.000000i 4.000000 + 1.000000i 0.000000 + 0.000000i
7.000000 - 5.000000i 1.000000 - 4.000000i 1.000000 + 0.000000i
Conjugate transpose:
2.000000 - 2.000000i 2.000000 + 1.000000i 7.000000 + 5.000000i
3.000000 - 1.000000i 4.000000 - 1.000000i 1.000000 + 4.000000i
-3.000000 - 5.000000i 0.000000 + 0.000000i 1.000000 + 0.000000i
Hermitian: false
Normal: false
Unitary: false
</pre>
=={{header|Common Lisp}}==
<syntaxhighlight lang="lisp">
(defun matrix-multiply (m1 m2)
(mapcar
(lambda (row)
(apply #'mapcar
(lambda (&rest column)
(apply #'+ (mapcar #'* row column))) m2)) m1))
(defun identity-p (m &optional (tolerance 1e-6))
"Is m an identity matrix?"
(loop for row in m
for r = 1 then (1+ r) do
(loop for col in row
for c = 1 then (1+ c) do
(if (eql r c)
(unless (< (abs (- col 1)) tolerance) (return-from identity-p nil))
(unless (< (abs col) tolerance) (return-from identity-p nil)) )))
T )
(defun conjugate-transpose (m)
(apply #'mapcar #'list (mapcar #'(lambda (r) (mapcar #'conjugate r)) m)) )
(defun hermitian-p (m)
(equalp m (conjugate-transpose m)))
(defun normal-p (m)
(let ((m* (conjugate-transpose m)))
(equalp (matrix-multiply m m*) (matrix-multiply m* m)) ))
(defun unitary-p (m)
(identity-p (matrix-multiply m (conjugate-transpose m))) )
</syntaxhighlight>
{{out}}
<pre>
(hermitian-p
'((3 #C(2 1))
(#C(2 -1) 1) ))
=> T
(normal-p
'((#C(0 1) 0)
(0 #C(3 -5)) ))
==> T
(unitary-p
'((0.70710677 0.70710677 0)
(#C(0 -0.70710677) #C(0 0.70710677) 0)
(0 0 1) ))
==> T
</pre>
=={{header|D}}==
{{trans|Python}} A well typed and mostly imperative version:
<syntaxhighlight lang="d">import std.stdio, std.complex, std.math, std.range, std.algorithm,
std.numeric;
T[][] conjugateTranspose(T)(in T[][] m) pure nothrow @safe {
auto r = new typeof(return)(m[0].length, m.length);
foreach (immutable nr, const row; m)
foreach (immutable nc, immutable c; row)
r[nc][nr] = c.conj;
return r;
}
bool isRectangular(T)(in T[][] M) pure nothrow @safe @nogc {
return M.all!(row => row.length == M[0].length);
}
T[][] matMul(T)(in T[][] A, in T[][] B) pure nothrow /*@safe*/
in {
assert(A.isRectangular && B.isRectangular &&
!A.empty && !B.empty && A[0].length == B.length);
} body {
auto result = new T[][](A.length, B[0].length);
auto aux = new T[B.length];
foreach (immutable j; 0 .. B[0].length) {
foreach (immutable k, const row; B)
aux[k] = row[j];
foreach (immutable i, const ai; A)
result[i][j] = dotProduct(ai, aux);
}
return result;
}
/// Check any number of complex matrices for equality within
/// some bits of mantissa.
bool areEqual(T)(in Complex!T[][][] matrices, in size_t nBits=20)
pure nothrow /*@safe*/ {
static bool allSame(U)(in U[] v) pure nothrow @nogc {
return v[1 .. $].all!(c => c == v[0]);
}
bool allNearSame(in Complex!T[] v) pure nothrow @nogc {
auto v0 = v[0].Complex!T; // To avoid another cast.
return v[1 .. $].all!(c => feqrel(v0.re, c.re) >= nBits &&
feqrel(v0.im, c.im) >= nBits);
}
immutable x = matrices.map!(m => m.length).array;
if (!allSame(x))
return false;
immutable y = matrices.map!(m => m[0].length).array;
if (!allSame(y))
return false;
foreach (immutable s; 0 .. x[0])
foreach (immutable t; 0 .. y[0])
if (!allNearSame(matrices.map!(m => m[s][t]).array))
return false;
return true;
}
bool isHermitian(T)(in Complex!T[][] m, in Complex!T[][] ct)
pure nothrow /*@safe*/ {
return [m, ct].areEqual;
}
bool isNormal(T)(in Complex!T[][] m, in Complex!T[][] ct)
pure nothrow /*@safe*/ {
return [matMul(m, ct), matMul(ct, m)].areEqual;
}
auto complexIdentitymatrix(in size_t side) pure nothrow /*@safe*/ {
return side.iota.map!(r => side.iota.map!(c => complex(r == c)).array).array;
}
bool isUnitary(T)(in Complex!T[][] m, in Complex!T[][] ct)
pure nothrow /*@safe*/ {
immutable mct = matMul(m, ct);
immutable ident = mct.length.complexIdentitymatrix;
return [mct, matMul(ct, m), ident].areEqual;
}
void main() /*@safe*/ {
alias C = complex;
immutable x = 2 ^^ 0.5 / 2;
immutable data = [[[C(3.0, 0.0), C(2.0, 1.0)],
[C(2.0, -1.0), C(1.0, 0.0)]],
[[C(1.0, 0.0), C(1.0, 0.0), C(0.0, 0.0)],
[C(0.0, 0.0), C(1.0, 0.0), C(1.0, 0.0)],
[C(1.0, 0.0), C(0.0, 0.0), C(1.0, 0.0)]],
[[C(x, 0.0), C(x, 0.0), C(0.0, 0.0)],
[C(0.0, -x), C(0.0, x), C(0.0, 0.0)],
[C(0.0, 0.0), C(0.0, 0.0), C(0.0, 1.0)]]];
foreach (immutable mat; data) {
enum mFormat = "[%([%(%1.3f, %)],\n %)]]";
writefln("Matrix:\n" ~ mFormat, mat);
immutable ct = conjugateTranspose(mat);
"Its conjugate transpose:".writeln;
writefln(mFormat, ct);
writefln("Hermitian? %s.", isHermitian(mat, ct));
writefln("Normal? %s.", isNormal(mat, ct));
writefln("Unitary? %s.\n", isUnitary(mat, ct));
}
}</syntaxhighlight>
{{out}}
<pre>Matrix:
[[3.000+0.000i, 2.000+1.000i],
[2.000-1.000i, 1.000+0.000i]]
Its conjugate transpose:
[[3.000-0.000i, 2.000+1.000i],
[2.000-1.000i, 1.000-0.000i]]
Hermitian? true.
Normal? true.
Unitary? false.
Matrix:
[[1.000+0.000i, 1.000+0.000i, 0.000+0.000i],
[0.000+0.000i, 1.000+0.000i, 1.000+0.000i],
[1.000+0.000i, 0.000+0.000i, 1.000+0.000i]]
Its conjugate transpose:
[[1.000-0.000i, 0.000-0.000i, 1.000-0.000i],
[1.000-0.000i, 1.000-0.000i, 0.000-0.000i],
[0.000-0.000i, 1.000-0.000i, 1.000-0.000i]]
Hermitian? false.
Normal? true.
Unitary? false.
Matrix:
[[0.707+0.000i, 0.707+0.000i, 0.000+0.000i],
[0.000-0.707i, 0.000+0.707i, 0.000+0.000i],
[0.000+0.000i, 0.000+0.000i, 0.000+1.000i]]
Its conjugate transpose:
[[0.707-0.000i, 0.000+0.707i, 0.000-0.000i],
[0.707-0.000i, 0.000-0.707i, 0.000-0.000i],
[0.000-0.000i, 0.000-0.000i, 0.000-1.000i]]
Hermitian? false.
Normal? true.
Unitary? true.
</pre>
===Alternative Version===
A more functional version that contains some typing problems (same output).
<syntaxhighlight lang="d">import std.stdio, std.complex, std.math, std.range, std.algorithm,
std.numeric, std.exception, std.traits;
// alias CM(T) = Complex!T[][]; // Not yet useful.
auto conjugateTranspose(T)(in Complex!T[][] m) pure nothrow /*@safe*/
if (!hasIndirections!T) {
return iota(m[0].length).map!(i => m.transversal(i).map!conj.array).array;
}
T[][] matMul(T)(immutable T[][] A, immutable T[][] B) pure nothrow /*@safe*/ {
immutable Bt = B[0].length.iota.map!(i => B.transversal(i).array).array;
return A.map!(a => Bt.map!(b => a.dotProduct(b)).array).array;
}
/// Check any number of complex matrices for equality within
/// some bits of mantissa.
bool areEqual(T)(in Complex!T[][][] matrices, in size_t nBits=20)
pure nothrow /*@safe*/ {
static bool allSame(U)(in U[] v) pure nothrow @nogc @safe {
return v[1 .. $].all!(c => c == v[0]);
}
bool allNearSame(in Complex!T[] v) pure nothrow @nogc @safe {
auto v0 = v[0].Complex!T; // To avoid another cast.
return v[1 .. $].all!(c => feqrel(v0.re, c.re) >= nBits &&
feqrel(v0.im, c.im) >= nBits);
}
immutable x = matrices.map!(m => m.length).array;
if (!allSame(x))
return false;
immutable y = matrices.map!(m => m[0].length).array;
if (!allSame(y))
return false;
foreach (immutable s; 0 .. x[0])
foreach (immutable t; 0 .. y[0])
if (!allNearSame(matrices.map!(m => m[s][t]).array))
return false;
return true;
}
bool isHermitian(T)(in Complex!T[][] m, in Complex!T[][] ct)
pure nothrow /*@safe*/ {
return [m, ct].areEqual;
}
bool isNormal(T)(immutable Complex!T[][] m, immutable Complex!T[][] ct)
pure nothrow /*@safe*/ {
return [matMul(m, ct), matMul(ct, m)].areEqual;
}
auto complexIdentitymatrix(in size_t side) pure nothrow /*@safe*/ {
return side.iota.map!(r => side.iota.map!(c => complex(r == c)).array).array;
}
bool isUnitary(T)(immutable Complex!T[][] m, immutable Complex!T[][] ct)
pure nothrow /*@safe*/ {
immutable mct = matMul(m, ct);
immutable ident = mct.length.complexIdentitymatrix;
return [mct, matMul(ct, m), ident].areEqual;
}
void main() {
alias C = complex;
immutable x = 2 ^^ 0.5 / 2;
foreach (/*immutable*/ const matrix;
[[[C(3.0, 0.0), C(2.0, 1.0)],
[C(2.0, -1.0), C(1.0, 0.0)]],
[[C(1.0, 0.0), C(1.0, 0.0), C(0.0, 0.0)],
[C(0.0, 0.0), C(1.0, 0.0), C(1.0, 0.0)],
[C(1.0, 0.0), C(0.0, 0.0), C(1.0, 0.0)]],
[[C(x, 0.0), C(x, 0.0), C(0.0, 0.0)],
[C(0.0, -x), C(0.0, x), C(0.0, 0.0)],
[C(0.0, 0.0), C(0.0, 0.0), C(0.0, 1.0)]]]) {
immutable mat = matrix.assumeUnique; //*
enum mFormat = "[%([%(%1.3f, %)],\n %)]]";
writefln("Matrix:\n" ~ mFormat, mat);
immutable ct = conjugateTranspose(mat);
"Its conjugate transpose:".writeln;
writefln(mFormat, ct);
writefln("Hermitian? %s.", isHermitian(mat, ct));
writefln("Normal? %s.", isNormal(mat, ct));
writefln("Unitary? %s.\n", isUnitary(mat, ct));
}
}</syntaxhighlight>
=={{header|F_Sharp|F#}}==
<syntaxhighlight lang="fsharp">
// Conjugate transpose. Nigel Galloway: January 10th., 2022
let fN g=let g=g|>List.map(List.map(fun(n,g)->System.Numerics.Complex(n,g)))|>MathNet.Numerics.LinearAlgebra.MatrixExtensions.matrix in (g,g.ConjugateTranspose())
let fG n g=(MathNet.Numerics.LinearAlgebra.Matrix.inverse n-g)|>MathNet.Numerics.LinearAlgebra.Matrix.forall(fun(n:System.Numerics.Complex)->abs n.Real<1e-14&&abs n.Imaginary<1e-14)
let test=[fN [[(3.0,0.0);(2.0,1.0)];[(2.0,-1.0);(1.0,0.0)]];fN [[(1.0,0.0);(1.0,0.0);(0.0,0.0)];[(0.0,0.0);(1.0,0.0);(1.0,0.0)];[(1.0,0.0);(0.0,0.0);(1.0,0.0)]];fN [[(1.0/sqrt 2.0,0.0);(1.0/sqrt 2.0,0.0);(0.0,0.0)];[(0.0,1.0/sqrt 2.0);(0.0,-1.0/sqrt 2.0);(0.0,0.0)];[(0.0,0.0);(0.0,0.0);(0.0,1.0)]]]
test|>List.iter(fun(n,g)->printfn $"Matrix\n------\n%A{n}\nConjugate transposed\n--------------------\n%A{g}\nIs hermitian: %A{n.IsHermitian()}\nIs normal: %A{n*g=g*n}\nIs unitary: %A{fG n g}\n")
</syntaxhighlight>
{{out}}
<pre>
Matrix
------
DenseMatrix 2x2-Complex
(3, 0) (2, 1)
(2, -1) (1, 0)
Conjugate transposed
--------------------
DenseMatrix 2x2-Complex
(3, -0) (2, 1)
(2, -1) (1, -0)
Is hermitian: true
Is normal: true
Is unitary: false
Matrix
------
DenseMatrix 3x3-Complex
(1, 0) (1, 0) (0, 0)
(0, 0) (1, 0) (1, 0)
(1, 0) (0, 0) (1, 0)
Conjugate transposed
--------------------
DenseMatrix 3x3-Complex
(1, -0) (0, -0) (1, -0)
(1, -0) (1, -0) (0, -0)
(0, -0) (1, -0) (1, -0)
Is hermitian: false
Is normal: true
Is unitary: false
Matrix
------
DenseMatrix 3x3-Complex
(0.707107, 0) (0.707107, 0) (0, 0)
(0, 0.707107) (0, -0.707107) (0, 0)
(0, 0) (0, 0) (0, 1)
Conjugate transposed
--------------------
DenseMatrix 3x3-Complex
(0.707107, -0) (0, -0.707107) (0, -0)
(0.707107, -0) (0, 0.707107) (0, -0)
(0, -0) (0, -0) (0, -1)
Is hermitian: false
Is normal: true
Is unitary: true
</pre>
=={{header|Factor}}==
Before the fix to [https://github.com/slavapestov/factor/issues/484 Factor bug #484], <code>m.</code> gave the wrong answer and this code failed. Factor 0.94 is too old to work.
{{works with|Factor|development (future 0.95)}}
<
IN: rosetta.hermitian
Line 322 ⟶ 1,201:
: unitary-matrix? ( matrix -- ? )
[ dup conj-t m. ] [ length identity-matrix ] bi = ;</
Usage:
Line 335 ⟶ 1,214:
t
f
=={{header|Fortran}}==
The examples and algorithms are taken from the j solution, except for UnitaryQ. The j solution uses the matrix inverse verb. Compilation on linux, assuming the program is file f.f08 :<pre>
gfortran -std=f2008 -Wall -fopenmp -ffree-form -fall-intrinsics -fimplicit-none f.f08 -o f</pre>
<syntaxhighlight lang="fortran">
program conjugate_transpose
complex, dimension(3, 3) :: a
integer :: i
a = reshape((/ (i, i=1,9) /), shape(a))
call characterize(a)
a(:2,:2) = reshape((/cmplx(3,0),cmplx(2,-1),cmplx(2,1),cmplx(1,0)/),(/2,2/))
call characterize(a(:2,:2))
call characterize(cmplx(reshape((/1,0,1,1,1,0,0,1,1/),(/3,3/)),0))
a(3,:) = (/cmplx(0,0), cmplx(0,0), cmplx(0,1)/)*sqrt(2.0)
a(2,:) = (/cmplx(0,-1),cmplx(0,1),cmplx(0,0)/)
a(1,:) = (/1,1,0/)
a = a * sqrt(2.0)/2.0
call characterize(a)
contains
subroutine characterize(a)
complex, dimension(:,:), intent(in) :: a
integer :: i, j
do i=1, size(a,1)
print *,(a(i, j), j=1,size(a,1))
end do
print *,'Is Hermitian? ',HermitianQ(a)
print *,'Is normal? ',NormalQ(a)
print *,'Unitary? ',UnitaryQ(a)
print '(/)'
end subroutine characterize
function ct(a) result(b) ! return the conjugate transpose of a matrix
complex, dimension(:,:), intent(in) :: a
complex, dimension(size(a,1),size(a,1)) :: b
b = conjg(transpose(a))
end function ct
function identity(n) result(b) ! return identity matrix
integer, intent(in) :: n
real, dimension(n,n) :: b
integer :: i
b = 0
do i=1, n
b(i,i) = 1
end do
end function identity
logical function HermitianQ(a)
complex, dimension(:,:), intent(in) :: a
HermitianQ = all(a .eq. ct(a))
end function HermitianQ
logical function NormalQ(a)
complex, dimension(:,:), intent(in) :: a
NormalQ = all(matmul(ct(a),a) .eq. matmul(a,ct(a)))
end function NormalQ
logical function UnitaryQ(a)
! if A inverse equals A star
! then multiplying each side by A should result in the identity matrix
! Thus show that A times A star is sufficiently close to I .
complex, dimension(:,:), intent(in) :: a
UnitaryQ = all(abs(matmul(a,ct(a)) - identity(size(a,1))) .lt. 1e-6)
end function UnitaryQ
end program conjugate_transpose
</syntaxhighlight>
<pre>
-*- mode: compilation; default-directory: "/tmp/" -*-
Compilation started at Fri Jun 7 16:31:38
a=./f && make $a && time $a
gfortran -std=f2008 -Wall -fopenmp -ffree-form -fall-intrinsics -fimplicit-none f.f08 -o f
( 1.00000000 , 0.00000000 ) ( 4.00000000 , 0.00000000 ) ( 7.00000000 , 0.00000000 )
( 2.00000000 , 0.00000000 ) ( 5.00000000 , 0.00000000 ) ( 8.00000000 , 0.00000000 )
( 3.00000000 , 0.00000000 ) ( 6.00000000 , 0.00000000 ) ( 9.00000000 , 0.00000000 )
Is Hermitian? F
Is normal? F
Unitary? F
( 3.00000000 , 0.00000000 ) ( 2.00000000 , 1.00000000 )
( 2.00000000 , -1.00000000 ) ( 1.00000000 , 0.00000000 )
Is Hermitian? T
Is normal? T
Unitary? F
( 1.00000000 , 0.00000000 ) ( 1.00000000 , 0.00000000 ) ( 0.00000000 , 0.00000000 )
( 0.00000000 , 0.00000000 ) ( 1.00000000 , 0.00000000 ) ( 1.00000000 , 0.00000000 )
( 1.00000000 , 0.00000000 ) ( 0.00000000 , 0.00000000 ) ( 1.00000000 , 0.00000000 )
Is Hermitian? F
Is normal? T
Unitary? F
( 0.707106769 , 0.00000000 ) ( 0.707106769 , 0.00000000 ) ( 0.00000000 , 0.00000000 )
( 0.00000000 ,-0.707106769 ) ( 0.00000000 , 0.707106769 ) ( 0.00000000 , 0.00000000 )
( 0.00000000 , 0.00000000 ) ( 0.00000000 , 0.00000000 ) ( 0.00000000 , 0.999999940 )
Is Hermitian? F
Is normal? T
Unitary? T
real 0m0.002s
user 0m0.000s
sys 0m0.000s
Compilation finished at Fri Jun 7 16:31:38
</pre>
=={{header|FreeBASIC}}==
<syntaxhighlight lang="freebasic">'complex type and operators for it
type complex
real as double
imag as double
end type
operator + ( a as complex, b as complex ) as complex
dim as complex r
r.real = a.real + b.real
r.imag = a.imag + b.imag
return r
end operator
operator * ( a as complex, b as complex ) as complex
dim as complex r
r.real = a.real*b.real - a.imag*b.imag
r.imag = a.real*b.imag + b.real*a.imag
return r
end operator
operator = ( a as complex, b as complex ) as boolean
if not a.real = b.real then return false
if not a.imag = b.imag then return false
return true
end operator
function complex_conjugate( a as complex ) as complex
dim as complex r
r.real = a.real
r.imag = -a.imag
return r
end function
'matrix type and operations for it
'reuses code from the matrix multiplication task
type Matrix
dim as complex m( any , any )
declare constructor ( )
declare constructor ( byval x as uinteger )
end type
constructor Matrix ( )
end constructor
constructor Matrix ( byval x as uinteger )
redim this.m( x - 1 , x - 1 )
end constructor
operator * ( byref a as Matrix , byref b as Matrix ) as Matrix
dim as Matrix ret
dim as uinteger i, j, k
redim ret.m( ubound( a.m , 1 ) , ubound( a.m , 1 ) )
for i = 0 to ubound( a.m , 1 )
for j = 0 to ubound( b.m , 2 )
for k = 0 to ubound( b.m , 1 )
ret.m( i , j ) += a.m( i , k ) * b.m( k , j )
next k
next j
next i
return ret
end operator
function conjugate_transpose( byref a as Matrix ) as Matrix
dim as Matrix ret
dim as uinteger i, j
redim ret.m( ubound( a.m , 1 ) , ubound( a.m , 1 ) )
for i = 0 to ubound( a.m , 1 )
for j = 0 to ubound( a.m , 2 )
ret.m( i, j ) = complex_conjugate(a.m( j, i ))
next j
next i
return ret
end function
'tests if matrices are unitary, hermitian, or normal
operator = (byref a as Matrix, byref b as matrix) as boolean
dim as integer i, j
if ubound(a.m, 1) <> ubound(b.m, 1) then return false
for i = 0 to ubound( a.m , 1 )
for j = 0 to ubound( a.m , 2 )
if not a.m(i,j)=b.m(i,j) then return false
next j
next i
return true
end operator
function is_identity( byref a as Matrix ) as boolean
dim as integer i, j
for i = 0 to ubound( a.m , 1 )
for j = 0 to ubound( a.m , 2 )
if i = j and ( not a.m(i,j).real = 1.0 or not a.m(i,j).imag = 0.0 ) then return false
if i <> j and ( not a.m(i,j).real = 0.0 or not a.m(i,j).imag = 0.0 ) then return false
next j
next i
return true
end function
function is_hermitian( byref a as Matrix ) as boolean
if a = conjugate_transpose(a) then return true
return false
end function
function is_normal( byref a as Matrix ) as boolean
dim as Matrix aa = conjugate_transpose(a)
if a*aa = aa*a then return true else return false
end function
function is_unitary( byref a as Matrix ) as boolean
dim as Matrix aa = conjugate_transpose(a)
if not is_identity( a*aa ) or not is_identity( aa*a ) then return false
return true
end function
'''now some example matrices
dim as Matrix A = Matrix(2) 'an identity matrix
A.m(0,0).real = 1.0 : A.m(0,0).imag = 0.0 : A.m(0,1).real = 0.0 : A.m(0,1).imag = 0.0
A.m(1,0).real = 0.0 : A.m(1,0).imag = 0.0 : A.m(1,1).real = 1.0 : A.m(1,1).imag = 0.0
dim as Matrix B = Matrix(2) 'a hermitian matrix
B.m(0,0).real = 1.0 : B.m(0,0).imag = 0.0 : B.m(0,1).real = 1.0 : B.m(0,1).imag = -1.0
B.m(1,0).real = 1.0 : B.m(1,0).imag = 1.0 : B.m(1,1).real = 1.0 : B.m(1,1).imag = 0.0
dim as Matrix C = Matrix(2) 'a random matrix
C.m(0,0).real = rnd : C.m(0,0).imag = rnd : C.m(0,1).real = rnd : C.m(0,1).imag = rnd
C.m(1,0).real = rnd : C.m(1,0).imag = rnd : C.m(1,1).real = rnd : C.m(1,1).imag = rnd
print is_hermitian(A), is_normal(A), is_unitary(A)
print is_hermitian(B), is_normal(B), is_unitary(B)
print is_hermitian(C), is_normal(C), is_unitary(C)</syntaxhighlight>
{{out}}
<pre>true true true
true true false
false false false</pre>
=={{header|Go}}==
<
import (
Line 448 ⟶ 1,578:
}
return m3
}</
Output:
<pre>
Line 488 ⟶ 1,618:
Unitary: true
</pre>
=={{header|Haskell}}==
Slow implementation using lists.
<syntaxhighlight lang="haskell">import Data.Complex (Complex(..), conjugate)
import Data.List (transpose)
type Matrix a = [[a]]
main :: IO ()
main =
mapM_
(\a -> do
putStrLn "\nMatrix:"
mapM_ print a
putStrLn "Conjugate Transpose:"
mapM_ print (conjTranspose a)
putStrLn $ "Hermitian? " ++ show (isHermitianMatrix a)
putStrLn $ "Normal? " ++ show (isNormalMatrix a)
putStrLn $ "Unitary? " ++ show (isUnitaryMatrix a))
([ [[3, 2 :+ 1], [2 :+ (-1), 1]]
, [[1, 1, 0], [0, 1, 1], [1, 0, 1]]
, [ [sqrt 2 / 2 :+ 0, sqrt 2 / 2 :+ 0, 0]
, [0 :+ sqrt 2 / 2, 0 :+ (-sqrt 2 / 2), 0]
, [0, 0, 0 :+ 1]
]
] :: [Matrix (Complex Double)])
isHermitianMatrix, isNormalMatrix, isUnitaryMatrix
:: RealFloat a
=> Matrix (Complex a) -> Bool
isHermitianMatrix = mTest id conjTranspose
isNormalMatrix = mTest mmct (mmul =<< conjTranspose)
isUnitaryMatrix = mTest mmct (ident . length)
mTest
:: RealFloat a
=> (a2 -> Matrix (Complex a)) -> (a2 -> Matrix (Complex a)) -> a2 -> Bool
mTest f g = (approxEqualMatrix . f) <*> g
mmct
:: RealFloat a
=> Matrix (Complex a) -> Matrix (Complex a)
mmct = mmul <*> conjTranspose
approxEqualMatrix
:: (Fractional a, Ord a)
=> Matrix (Complex a) -> Matrix (Complex a) -> Bool
approxEqualMatrix a b =
length a == length b &&
length (head a) == length (head b) &&
and (zipWith approxEqualComplex (concat a) (concat b))
where
approxEqualComplex (rx :+ ix) (ry :+ iy) =
abs (rx - ry) < eps && abs (ix - iy) < eps
eps = 1e-14
mmul
:: Num a
=> Matrix a -> Matrix a -> Matrix a
mmul a b =
[ [ sum (zipWith (*) row column)
| column <- transpose b ]
| row <- a ]
ident
:: Num a
=> Int -> Matrix a
ident size =
[ [ fromIntegral $ div a b * div b a
| a <- [1 .. size] ]
| b <- [1 .. size] ]
conjTranspose
:: Num a
=> Matrix (Complex a) -> Matrix (Complex a)
conjTranspose = map (map conjugate) . transpose</syntaxhighlight>
Output:
<pre>
Matrix:
[3.0 :+ 0.0,2.0 :+ 1.0]
[2.0 :+ (-1.0),1.0 :+ 0.0]
Conjugate Transpose:
[3.0 :+ (-0.0),2.0 :+ 1.0]
[2.0 :+ (-1.0),1.0 :+ (-0.0)]
Hermitian? True
Normal? True
Unitary? False
Matrix:
[1.0 :+ 0.0,1.0 :+ 0.0,0.0 :+ 0.0]
[0.0 :+ 0.0,1.0 :+ 0.0,1.0 :+ 0.0]
[1.0 :+ 0.0,0.0 :+ 0.0,1.0 :+ 0.0]
Conjugate Transpose:
[1.0 :+ (-0.0),0.0 :+ (-0.0),1.0 :+ (-0.0)]
[1.0 :+ (-0.0),1.0 :+ (-0.0),0.0 :+ (-0.0)]
[0.0 :+ (-0.0),1.0 :+ (-0.0),1.0 :+ (-0.0)]
Hermitian? False
Normal? True
Unitary? False
Matrix:
[0.7071067811865476 :+ 0.0,0.7071067811865476 :+ 0.0,0.0 :+ 0.0]
[0.0 :+ 0.7071067811865476,0.0 :+ (-0.7071067811865476),0.0 :+ 0.0]
[0.0 :+ 0.0,0.0 :+ 0.0,0.0 :+ 1.0]
Conjugate Transpose:
[0.7071067811865476 :+ (-0.0),0.0 :+ (-0.7071067811865476),0.0 :+ (-0.0)]
[0.7071067811865476 :+ (-0.0),0.0 :+ 0.7071067811865476,0.0 :+ (-0.0)]
[0.0 :+ (-0.0),0.0 :+ (-0.0),0.0 :+ (-1.0)]
Hermitian? False
Normal? True
Unitary? True
</pre>
=={{header|J}}==
'''
<code>+</code> when used without a left argument is conjugate, <code>|:</code> is transpose, and <code>@</code> composes functions.
'''Examples''': <syntaxhighlight lang="j"> X =: +/ . * NB. Matrix Multiply (x)
HERMITIAN =: 3 2j1 ,: 2j_1 1
Line 503 ⟶ 1,751:
UNITARY =: (-:%:2) * 1 1 0 , 0j_1 0j1 0 ,: 0 0 0j1 * %:2
(ct -: %.) UNITARY NB. A_ct = A^-1
1</
'''Reference''' (with example matrices
+--------+-----+--------------------------+
| 3 2j1|1 1 0| 0.707107 0.707107 0|
Line 511 ⟶ 1,759:
| |1 0 1| 0 0 0j1|
+--------+-----+--------------------------+
NB. In J, PjQ is P + Q*i and the 0.7071... is sqrt(2)
hermitian=: -: ct
normal =: (X~ -: X) ct
unitary=: ct -: %.
(hermitian,normal,unitary)&.>HERMITIAN;NORMAL;UNITARY
+-----+-----+-----+
|1 1 0|0 1 0|0 1 1|
+-----+-----+-----+</syntaxhighlight>
=={{header|Java}}==
<syntaxhighlight lang ="java">
import java.util.Arrays;
import java.util.List;
public final class ConjugateTranspose {
public static void main(String[] aArgs) {
ComplexMatrix one = new ComplexMatrix( new Complex[][] { { new Complex(0, 4), new Complex(-1, 1) },
{ new Complex(1, -1), new Complex(0, 4) } } );
ComplexMatrix two = new ComplexMatrix(
new Complex[][] { { new Complex(1, 0), new Complex(1, 1), new Complex(0, 2) },
{ new Complex(1, -1), new Complex(5, 0), new Complex(-3, 0) },
{ new Complex(0, -2), new Complex(-3, 0), new Complex(0, 0) } } );
final double term = 1.0 / Math.sqrt(2.0);
ComplexMatrix three = new ComplexMatrix( new Complex[][] { { new Complex(term, 0), new Complex(term, 0) },
{ new Complex(0, term), new Complex(0, -term) } } );
List<ComplexMatrix> matricies = List.of( one, two, three );
for ( ComplexMatrix matrix : matricies ) {
System.out.println("Matrix:");
matrix.display();
System.out.println("Conjugate transpose:");
matrix.conjugateTranspose().display();
System.out.println("Hermitian: " + matrix.isHermitian());
System.out.println("Normal: " + matrix.isNormal());
System.out.println("Unitary: " + matrix.isUnitary() + System.lineSeparator());
}
}
}
final class ComplexMatrix {
public ComplexMatrix(Complex[][] aData) {
rowCount = aData.length;
colCount = aData[0].length;
data = Arrays.stream(aData).map( row -> Arrays.copyOf(row, row.length) ).toArray(Complex[][]::new);
}
public ComplexMatrix multiply(ComplexMatrix aOther) {
if ( colCount != aOther.rowCount ) {
throw new RuntimeException("Incompatible matrix dimensions.");
}
Complex[][] newData = new Complex[rowCount][aOther.colCount];
Arrays.stream(newData).forEach( row -> Arrays.fill(row, new Complex(0, 0)) );
for ( int row = 0; row < rowCount; row++ ) {
for ( int col = 0; col < aOther.colCount; col++ ) {
for ( int k = 0; k < colCount; k++ ) {
newData[row][col] = newData[row][col].add(data[row][k].multiply(aOther.data[k][col]));
}
}
}
return new ComplexMatrix(newData);
}
public ComplexMatrix conjugateTranspose() {
if ( rowCount != colCount ) {
throw new IllegalArgumentException("Only applicable to a square matrix");
}
Complex[][] newData = new Complex[colCount][rowCount];
for ( int row = 0; row < rowCount; row++ ) {
for ( int col = 0; col < colCount; col++ ) {
newData[col][row] = data[row][col].conjugate();
}
}
return new ComplexMatrix(newData);
}
public static ComplexMatrix identity(int aSize) {
Complex[][] data = new Complex[aSize][aSize];
for ( int row = 0; row < aSize; row++ ) {
for ( int col = 0; col < aSize; col++ ) {
data[row][col] = ( row == col ) ? new Complex(1, 0) : new Complex(0, 0);
}
}
return new ComplexMatrix(data);
}
public boolean equals(ComplexMatrix aOther) {
if ( aOther.rowCount != rowCount || aOther.colCount != colCount ) {
return false;
}
for ( int row = 0; row < rowCount; row++ ) {
for ( int col = 0; col < colCount; col++ ) {
if ( data[row][col].subtract(aOther.data[row][col]).modulus() > EPSILON ) {
return false;
}
}
}
return true;
}
public void display() {
for ( int row = 0; row < rowCount; row++ ) {
System.out.print("[");
for ( int col = 0; col < colCount - 1; col++ ) {
System.out.print(data[row][col] + ", ");
}
System.out.println(data[row][colCount - 1] + " ]");
}
}
public boolean isHermitian() {
return equals(conjugateTranspose());
}
public boolean isNormal() {
ComplexMatrix conjugateTranspose = conjugateTranspose();
return multiply(conjugateTranspose).equals(conjugateTranspose.multiply(this));
}
public boolean isUnitary() {
ComplexMatrix conjugateTranspose = conjugateTranspose();
return multiply(conjugateTranspose).equals(identity(rowCount)) &&
conjugateTranspose.multiply(this).equals(identity(rowCount));
}
private final int rowCount;
private final int colCount;
private final Complex[][] data;
private static final double EPSILON = 0.000_000_000_001;
}
final class Complex {
public Complex(double aReal, double aImag) {
real = aReal;
imag = aImag;
}
public Complex add(Complex aOther) {
return new Complex(real + aOther.real, imag + aOther.imag);
}
public Complex multiply(Complex aOther) {
return new Complex(real * aOther.real - imag * aOther.imag, real * aOther.imag + imag * aOther.real);
}
public Complex negate() {
return new Complex(-real, -imag);
}
public Complex subtract(Complex aOther) {
return this.add(aOther.negate());
}
public Complex conjugate() {
return new Complex(real, -imag);
}
public double modulus() {
return Math.hypot(real, imag);
}
public boolean equals(Complex aOther) {
return real == aOther.real && imag == aOther.imag;
}
@Override
public String toString() {
String prefix = ( real < 0.0 ) ? "" : " ";
String realPart = prefix + String.format("%.3f", real);
String sign = ( imag < 0.0 ) ? " - " : " + ";
return realPart + sign + String.format("%.3f", Math.abs(imag)) + "i";
}
private final double real;
private final double imag;
}
</syntaxhighlight>
{{ out }}
<pre>
Matrix:
[ 0.000 + 4.000i, -1.000 + 1.000i ]
[ 1.000 - 1.000i, 0.000 + 4.000i ]
Conjugate transpose:
[ 0.000 - 4.000i, 1.000 + 1.000i ]
[-1.000 - 1.000i, 0.000 - 4.000i ]
Hermitian: false
Normal: true
Unitary: false
Matrix:
[ 1.000 + 0.000i, 1.000 + 1.000i, 0.000 + 2.000i ]
[ 1.000 - 1.000i, 5.000 + 0.000i, -3.000 + 0.000i ]
[ 0.000 - 2.000i, -3.000 + 0.000i, 0.000 + 0.000i ]
Conjugate transpose:
[ 1.000 + 0.000i, 1.000 + 1.000i, 0.000 + 2.000i ]
[ 1.000 - 1.000i, 5.000 + 0.000i, -3.000 + 0.000i ]
[ 0.000 - 2.000i, -3.000 + 0.000i, 0.000 + 0.000i ]
Hermitian: true
Normal: true
Unitary: false
Matrix:
[ 0.707 + 0.000i, 0.707 + 0.000i ]
[ 0.000 + 0.707i, 0.000 - 0.707i ]
Conjugate transpose:
[ 0.707 + 0.000i, 0.000 - 0.707i ]
[ 0.707 + 0.000i, 0.000 + 0.707i ]
Hermitian: false
Normal: true
Unitary: true
</pre>
=={{header|jq}}==
{{works with|jq|1.4}}
In the following, we use the array [x,y] to represent the complex number x + iy,
but the following functions also accept a number wherever a complex number is acceptable.
====Infrastructure====
'''(1) transpose/0''':
If your jq does not have "transpose" then the following may be used:
<syntaxhighlight lang="jq"># transpose/0 expects its input to be a rectangular matrix
# (an array of equal-length arrays):
def transpose:
if (.[0] | length) == 0 then []
else [map(.[0])] + (map(.[1:]) | transpose)
end ;</syntaxhighlight>
'''(2) Operations on real/complex numbers'''
<syntaxhighlight lang="jq"># x must be real or complex, and ditto for y;
# always return complex
def plus(x; y):
if (x|type) == "number" then
if (y|type) == "number" then [ x+y, 0 ]
else [ x + y[0], y[1]]
end
elif (y|type) == "number" then plus(y;x)
else [ x[0] + y[0], x[1] + y[1] ]
end;
# x must be real or complex, and ditto for y;
# always return complex
def multiply(x; y):
if (x|type) == "number" then
if (y|type) == "number" then [ x*y, 0 ]
else [x * y[0], x * y[1]]
end
elif (y|type) == "number" then multiply(y;x)
else [ x[0] * y[0] - x[1] * y[1], x[0] * y[1] + x[1] * y[0]]
end;
# conjugate of a real or complex number
def conjugate:
if type == "number" then [.,0]
else [.[0], -(.[1]) ]
end;</syntaxhighlight>
'''(3) Array operations'''
<syntaxhighlight lang="jq"># a and b are arrays of real/complex numbers
def dot_product(a; b):
a as $a | b as $b
| reduce range(0;$a|length) as $i
(0; . as $s | plus($s; multiply($a[$i]; $b[$i]) ));</syntaxhighlight>
'''(4) Matrix operations'''
<syntaxhighlight lang="jq"># convert a matrix of mixed real/complex entries to all complex entries
def to_complex:
def toc: if type == "number" then [.,0] else . end;
map( map(toc) );
# simple matrix pretty-printer
def pp(wide):
def pad: tostring | (wide - length) * " " + .;
def row: reduce .[] as $x (""; . + ($x|pad));
reduce .[] as $row (""; . + "\n\($row|row)");
# Matrix multiplication
# A and B should both be real/complex matrices,
# A being m by n, and B being n by p.
def matrix_multiply(A; B):
A as $A | B as $B
| ($B[0]|length) as $p
| ($B|transpose) as $BT
| reduce range(0; $A|length) as $i
([]; reduce range(0; $p) as $j
(.; .[$i][$j] = dot_product( $A[$i]; $BT[$j] ) )) ;
# Complex identity matrix of dimension n
def complex_identity(n):
def indicator(i;n): [range(0;n)] | map( [0,0]) | .[i] = [1,0];
reduce range(0; n) as $i ([]; . + [indicator( $i; n )] );
# Approximate equality of two matrices
# Are two real/complex matrices essentially equal
# in the sense that the sum of the squared element-wise differences
# is less than or equal to epsilon?
# The two matrices must be conformal.
def approximately_equal(M; N; epsilon):
def norm: multiply(. ; conjugate ) | .[0];
def sqdiff( x; y): plus(x; multiply(y; -1)) | norm;
reduce range(0;M|length) as $i
(0; reduce range(0; M[0]|length) as $j
(.; 0 + sqdiff( M[$i][$j]; N[$i][$j] ) ) ) <= epsilon;</syntaxhighlight>
====Conjugate transposition====
<syntaxhighlight lang="jq"># (entries may be real and/or complex)
def conjugate_transpose:
map( map(conjugate) ) | transpose;
# A Hermitian matrix equals its own conjugate transpose
def is_hermitian:
to_complex == conjugate_transpose;
# A matrix is normal if it commutes multiplicatively
# with its conjugate transpose
def is_normal:
. as $M
| conjugate_transpose as $H
| matrix_multiply($H; $M) == matrix_multiply($H; $M);
# A unitary matrix (U) has its inverse equal to its conjugate transpose (T)
# i.e. U^-1 == T; NASC is I == UT == TU
def is_unitary:
. as $M
| conjugate_transpose as $H
| complex_identity(length) as $I
| approximately_equal( $I; matrix_multiply($H;$M); 1e-10)
and approximately_equal( $I ; matrix_multiply($M;$H); 1e-10) ; </syntaxhighlight>
====Examples====
<syntaxhighlight lang="jq">def hermitian_example:
[ [ 3, [2,1]],
[[2,-1], 1 ] ];
def normal_example:
[ [1, 1, 0],
[0, 1, 1],
[1, 0, 1] ];
def unitary_example:
0.707107
| [ [ [., 0], [., 0], 0 ],
[ [0, -.], [0, .], 0 ],
[ 0, 0, [0,1] ] ];
def demo:
hermitian_example
| ("Hermitian example:", pp(8)),
"",
("Its conjugate transpose is:", (to_complex | conjugate_transpose | pp(8))),
"",
"Hermitian example: \(hermitian_example | is_hermitian )",
"",
"Normal example: \(normal_example | is_normal )",
"",
"Unitary example: \(unitary_example | is_unitary)"
;
demo</syntaxhighlight>
{{out}}
<syntaxhighlight lang="sh">$ jq -r -c -n -f Conjugate_transpose.jq
Hermitian example:
3 [2,1]
[2,-1] 1
Conjugate transpose:
[3,-0] [2,1]
[2,-1] [1,-0]
Hermitian example: true
Normal example: true
Unitary example: true</syntaxhighlight>
=={{header|Julia}}==
Julia has a built-in matrix type, and the conjugate-transpose of a complex matrix <code>A</code> is simply:
<syntaxhighlight lang="julia">A'</syntaxhighlight>
(similar to Matlab). You can check whether <code>A</code> is Hermitian via the built-in function
<syntaxhighlight lang="julia">ishermitian(A)</syntaxhighlight>
Ignoring the possibility of roundoff errors for floating-point matrices (like most of the examples in the other languages), you can check whether a matrix is normal or unitary by the following functions
<syntaxhighlight lang="julia">eye(A) = A^0
isnormal(A) = size(A,1) == size(A,2) && A'*A == A*A'
isunitary(A) = size(A,1) == size(A,2) && A'*A == eye(A)</syntaxhighlight>
=={{header|Kotlin}}==
As Kotlin doesn't have built in classes for complex numbers or matrices, some basic functionality needs to be coded in order to tackle this task:
<syntaxhighlight lang="scala">// version 1.1.3
typealias C = Complex
typealias Vector = Array<C>
typealias Matrix = Array<Vector>
class Complex(val real: Double, val imag: Double) {
operator fun plus(other: Complex) =
Complex(this.real + other.real, this.imag + other.imag)
operator fun times(other: Complex) =
Complex(this.real * other.real - this.imag * other.imag,
this.real * other.imag + this.imag * other.real)
fun conj() = Complex(this.real, -this.imag)
/* tolerable equality allowing for rounding of Doubles */
infix fun teq(other: Complex) =
Math.abs(this.real - other.real) <= 1e-14 &&
Math.abs(this.imag - other.imag) <= 1e-14
override fun toString() = "${"%.3f".format(real)} " + when {
imag > 0.0 -> "+ ${"%.3f".format(imag)}i"
imag == 0.0 -> "+ 0.000i"
else -> "- ${"%.3f".format(-imag)}i"
}
}
fun Matrix.conjTranspose(): Matrix {
val rows = this.size
val cols = this[0].size
return Matrix(cols) { i -> Vector(rows) { j -> this[j][i].conj() } }
}
operator fun Matrix.times(other: Matrix): Matrix {
val rows1 = this.size
val cols1 = this[0].size
val rows2 = other.size
val cols2 = other[0].size
require(cols1 == rows2)
val result = Matrix(rows1) { Vector(cols2) { C(0.0, 0.0) } }
for (i in 0 until rows1) {
for (j in 0 until cols2) {
for (k in 0 until rows2) {
result[i][j] += this[i][k] * other[k][j]
}
}
}
return result
}
/* tolerable matrix equality using the same concept as for complex numbers */
infix fun Matrix.teq(other: Matrix): Boolean {
if (this.size != other.size || this[0].size != other[0].size) return false
for (i in 0 until this.size) {
for (j in 0 until this[0].size) if (!(this[i][j] teq other[i][j])) return false
}
return true
}
fun Matrix.isHermitian() = this teq this.conjTranspose()
fun Matrix.isNormal(): Boolean {
val ct = this.conjTranspose()
return (this * ct) teq (ct * this)
}
fun Matrix.isUnitary(): Boolean {
val ct = this.conjTranspose()
val prod = this * ct
val ident = identityMatrix(prod.size)
val prod2 = ct * this
return (prod teq ident) && (prod2 teq ident)
}
fun Matrix.print() {
val rows = this.size
val cols = this[0].size
for (i in 0 until rows) {
for (j in 0 until cols) {
print(this[i][j])
print(if(j < cols - 1) ", " else "\n")
}
}
println()
}
fun identityMatrix(n: Int): Matrix {
require(n >= 1)
val ident = Matrix(n) { Vector(n) { C(0.0, 0.0) } }
for (i in 0 until n) ident[i][i] = C(1.0, 0.0)
return ident
}
fun main(args: Array<String>) {
val x = Math.sqrt(2.0) / 2.0
val matrices = arrayOf(
arrayOf(
arrayOf(C(3.0, 0.0), C(2.0, 1.0)),
arrayOf(C(2.0, -1.0), C(1.0, 0.0))
),
arrayOf(
arrayOf(C(1.0, 0.0), C(1.0, 0.0), C(0.0, 0.0)),
arrayOf(C(0.0, 0.0), C(1.0, 0.0), C(1.0, 0.0)),
arrayOf(C(1.0, 0.0), C(0.0, 0.0), C(1.0, 0.0))
),
arrayOf(
arrayOf(C(x, 0.0), C(x, 0.0), C(0.0, 0.0)),
arrayOf(C(0.0, -x), C(0.0, x), C(0.0, 0.0)),
arrayOf(C(0.0, 0.0), C(0.0, 0.0), C(0.0, 1.0))
)
)
for (m in matrices) {
println("Matrix:")
m.print()
val mct = m.conjTranspose()
println("Conjugate transpose:")
mct.print()
println("Hermitian? ${mct.isHermitian()}")
println("Normal? ${mct.isNormal()}")
println("Unitary? ${mct.isUnitary()}\n")
}
}</syntaxhighlight>
{{out}}
<pre>
Matrix:
3.000 + 0.000i, 2.000 + 1.000i
2.000 - 1.000i, 1.000 + 0.000i
Conjugate transpose:
3.000 + 0.000i, 2.000 + 1.000i
2.000 - 1.000i, 1.000 + 0.000i
Hermitian? true
Normal? true
Unitary? false
Matrix:
1.000 + 0.000i, 1.000 + 0.000i, 0.000 + 0.000i
0.000 + 0.000i, 1.000 + 0.000i, 1.000 + 0.000i
1.000 + 0.000i, 0.000 + 0.000i, 1.000 + 0.000i
Conjugate transpose:
1.000 + 0.000i, 0.000 + 0.000i, 1.000 + 0.000i
1.000 + 0.000i, 1.000 + 0.000i, 0.000 + 0.000i
0.000 + 0.000i, 1.000 + 0.000i, 1.000 + 0.000i
Hermitian? false
Normal? true
Unitary? false
Matrix:
0.707 + 0.000i, 0.707 + 0.000i, 0.000 + 0.000i
0.000 - 0.707i, 0.000 + 0.707i, 0.000 + 0.000i
0.000 + 0.000i, 0.000 + 0.000i, 0.000 + 1.000i
Conjugate transpose:
0.707 + 0.000i, 0.000 + 0.707i, 0.000 + 0.000i
0.707 + 0.000i, 0.000 - 0.707i, 0.000 + 0.000i
0.000 + 0.000i, 0.000 + 0.000i, 0.000 - 1.000i
Hermitian? false
Normal? true
Unitary? true
</pre>
=={{header|Maple}}==
The commands <code>HermitianTranspose</code> and <code>IsUnitary</code> are provided by the <code>LinearAlgebra</code> package.
<syntaxhighlight lang="maple">M:=<<3|2+I>,<2-I|1>>:
with(LinearAlgebra):
IsNormal:=A->EqualEntries(A^%H.A,A.A^%H):
M^%H;
HermitianTranspose(M);
type(M,'Matrix'(hermitian));
IsNormal(M);
IsUnitary(M);</syntaxhighlight>
Output:
<pre> [ 3 2 + I]
[ ]
[2 - I 1 ]
[ 3 2 + I]
[ ]
[2 - I 1 ]
true
true
false</pre>
=={{header|Mathematica}} / {{header|Wolfram Language}}==
<syntaxhighlight lang="mathematica">NormalMatrixQ[a_List?MatrixQ] := Module[{b = Conjugate@Transpose@a},a.b === b.a]
UnitaryQ[m_List?MatrixQ] := (Conjugate@Transpose@m.m == IdentityMatrix@Length@m)
Line 530 ⟶ 2,369:
{HermitianMatrixQ@#, NormalMatrixQ@#, UnitaryQ@#}&@m
-> {False, False, False}</
=={{header|Nim}}==
The complex type is defined as generic regarding the type of real an imaginary part. We have chosen to use Complex[float] and make only our Matrix type generic regarding the dimensions. Thus, a Matrix has a two dimensions M and N which are static, i.e. known at compile time. We have enforced the condition M = N for square matrices (also at compile time).
<syntaxhighlight lang="nim">import complex, strformat
type Matrix[M, N: static Positive] = array[M, array[N, Complex[float]]]
const Eps = 1e-10 # Tolerance used for float comparisons.
####################################################################################################
# Templates.
template `[]`(m: Matrix; i, j: Natural): Complex[float] =
## Allow to get value of an element using m[i, j] syntax.
m[i][j]
template `[]=`(m: var Matrix; i, j: Natural; val: Complex[float]) =
## Allow to set value of an element using m[i, j] syntax.
m[i][j] = val
####################################################################################################
# General operations.
func `$`(m: Matrix): string =
## Return the string representation of a matrix using one line per row.
for i, row in m:
result.add(if i == 0: '[' else: ' ')
for j, val in row:
if j != 0: result.add(' ')
result.add(&"({val.re:7.4f}, {val.im:7.4f})")
result.add(if i == m.high: ']' else: '\n')
#---------------------------------------------------------------------------------------------------
func conjugateTransposed[M, N: static int](m: Matrix[M, N]): Matrix[N, M] =
## Return the conjugate transpose of a matrix.
for i in 0..<m.M:
for j in 0..<m.N:
result[j, i] = m[i, j].conjugate()
#---------------------------------------------------------------------------------------------------
func `*`[M, K, N: static int](m1: Matrix[M, K]; m2: Matrix[K, N]): Matrix[M, N] =
# Compute the product of two matrices.
for i in 0..<M:
for j in 0..<N:
for k in 0..<K:
result[i, j] = result[i, j] + m1[i, k] * m2[k, j]
####################################################################################################
# Properties.
func isHermitian(m: Matrix): bool =
## Check if a matrix is hermitian.
when m.M != m.N:
{.error: "hermitian test only allowed for square matrices".}
else:
for i in 0..<m.M:
for j in i..<m.N:
if m[i, j] != m[j, i].conjugate:
return false
result = true
#---------------------------------------------------------------------------------------------------
func isNormal(m: Matrix): bool =
## Check if a matrix is normal.
when m.M != m.N:
{.error: "normal test only allowed for square matrices".}
else:
let h = m.conjugateTransposed
result = m * h == h * m
#---------------------------------------------------------------------------------------------------
func isIdentity(m: Matrix): bool =
## Check if a matrix is the identity matrix.
when m.M != m.N:
{.error: "identity test only allowed for square matrices".}
else:
for i in 0..<m.M:
for j in 0..<m.N:
if i == j:
if abs(m[i, j] - 1.0) > Eps:
return false
else:
if abs(m[i, j]) > Eps:
return false
result = true
#---------------------------------------------------------------------------------------------------
func isUnitary(m: Matrix): bool =
## Check if a matrix is unitary.
when m.M != m.N:
{.error: "unitary test only allowed for square matrices".}
else:
let h = m.conjugateTransposed
result = (m * h).isIdentity and (h * m).isIdentity
#———————————————————————————————————————————————————————————————————————————————————————————————————
when isMainModule:
import math
proc test(m: Matrix) =
echo "\n"
echo "Matrix"
echo "------"
echo m
echo ""
echo "Conjugate transposed"
echo "--------------------"
echo m.conjugateTransposed
when m.M == m.N:
# Only for squares matrices.
echo ""
echo "Hermitian: ", m.isHermitian
echo "Normal: ", m.isNormal
echo "Unitary: ", m.isUnitary
#-------------------------------------------------------------------------------------------------
# Non square matrix.
const M1: Matrix[2, 3] = [[1.0 + im 2.0, 3.0 + im 0.0, 2.0 + im 5.0],
[3.0 - im 1.0, 2.0 + im 0.0, 0.0 + im 3.0]]
# Square matrices.
const M2: Matrix[2, 2] = [[3.0 + im 0.0, 2.0 + im 1.0],
[2.0 - im 1.0, 1.0 + im 0.0]]
const M3: Matrix[3, 3] = [[1.0 + im 0.0, 1.0 + im 0.0, 0.0 + im 0.0],
[0.0 + im 0.0, 1.0 + im 0.0, 1.0 + im 0.0],
[1.0 + im 0.0, 0.0 + im 0.0, 1.0 + im 0.0]]
const SR2 = 1 / sqrt(2.0)
const M4: Matrix[3, 3] = [[SR2 + im 0.0, SR2 + im 0.0, 0.0 + im 0.0],
[0.0 + im SR2, 0.0 - im SR2, 0.0 + im 0.0],
[0.0 + im 0.0, 0.0 + im 0.0, 0.0 + im 1.0]]
test(M1)
test(M2)
test(M3)
test(M4)</syntaxhighlight>
{{out}}
<pre>Matrix
------
[( 1.0000, 2.0000) ( 3.0000, 0.0000) ( 2.0000, 5.0000)
( 3.0000, -1.0000) ( 2.0000, 0.0000) ( 0.0000, 3.0000)]
Conjugate transposed
--------------------
[( 1.0000, -2.0000) ( 3.0000, 1.0000)
( 3.0000, -0.0000) ( 2.0000, -0.0000)
( 2.0000, -5.0000) ( 0.0000, -3.0000)]
Matrix
------
[( 3.0000, 0.0000) ( 2.0000, 1.0000)
( 2.0000, -1.0000) ( 1.0000, 0.0000)]
Conjugate transposed
--------------------
[( 3.0000, -0.0000) ( 2.0000, 1.0000)
( 2.0000, -1.0000) ( 1.0000, -0.0000)]
Hermitian: true
Normal: true
Unitary: false
Matrix
------
[( 1.0000, 0.0000) ( 1.0000, 0.0000) ( 0.0000, 0.0000)
( 0.0000, 0.0000) ( 1.0000, 0.0000) ( 1.0000, 0.0000)
( 1.0000, 0.0000) ( 0.0000, 0.0000) ( 1.0000, 0.0000)]
Conjugate transposed
--------------------
[( 1.0000, -0.0000) ( 0.0000, -0.0000) ( 1.0000, -0.0000)
( 1.0000, -0.0000) ( 1.0000, -0.0000) ( 0.0000, -0.0000)
( 0.0000, -0.0000) ( 1.0000, -0.0000) ( 1.0000, -0.0000)]
Hermitian: false
Normal: true
Unitary: false
Matrix
------
[( 0.7071, 0.0000) ( 0.7071, 0.0000) ( 0.0000, 0.0000)
( 0.0000, 0.7071) ( 0.0000, -0.7071) ( 0.0000, 0.0000)
( 0.0000, 0.0000) ( 0.0000, 0.0000) ( 0.0000, 1.0000)]
Conjugate transposed
--------------------
[( 0.7071, -0.0000) ( 0.0000, -0.7071) ( 0.0000, -0.0000)
( 0.7071, -0.0000) ( 0.0000, 0.7071) ( 0.0000, -0.0000)
( 0.0000, -0.0000) ( 0.0000, -0.0000) ( 0.0000, -1.0000)]
Hermitian: false
Normal: true
Unitary: true</pre>
=={{header|PARI/GP}}==
<syntaxhighlight lang="text">conjtranspose(M)=conj(M~)
isHermitian(M)=M==conj(M~)
isnormal(M)=my(H=conj(M~));H*M==M*H
isunitary(M)=M*conj(M~)==1</syntaxhighlight>
=={{header|Perl}}==
In general, using two or more modules which overload operators can be problematic. For this task, using both Math::Complex and Math::MatrixReal gives us the behavior we want for everything except matrix I/O, i.e. parsing and stringification.
<syntaxhighlight lang="perl">use strict;
use warnings;
use English;
use Math::Complex;
use Math::MatrixReal;
my @examples = (example1(), example2(), example3());
foreach my $m (@examples) {
print "Starting matrix:\n", cmat_as_string($m), "\n";
my $m_ct = conjugate_transpose($m);
print "Its conjugate transpose:\n", cmat_as_string($m_ct), "\n";
print "Is Hermitian? ", (cmats_are_equal($m, $m_ct) ? 'TRUE' : 'FALSE'), "\n";
my $product = $m_ct * $m;
print "Is normal? ", (cmats_are_equal($product, $m * $m_ct) ? 'TRUE' : 'FALSE'), "\n";
my $I = identity(($m->dim())[0]);
print "Is unitary? ", (cmats_are_equal($product, $I) ? 'TRUE' : 'FALSE'), "\n";
print "\n";
}
exit 0;
sub cmats_are_equal {
my ($m1, $m2) = @ARG;
my $max_norm = 1.0e-7;
return abs($m1 - $m2) < $max_norm; # Math::MatrixReal overloads abs().
}
# Note that Math::Complex and Math::MatrixReal both overload '~', for
# complex conjugates and matrix transpositions respectively.
sub conjugate_transpose {
my $m_T = ~ shift;
my $result = $m_T->each(sub {~ $ARG[0]});
return $result;
}
sub cmat_as_string {
my $m = shift;
my $n_rows = ($m->dim())[0];
my @row_strings = map { q{[} . join(q{, }, $m->row($ARG)->as_list) . q{]} }
(1 .. $n_rows);
return join("\n", @row_strings);
}
sub identity {
my $N = shift;
my $m = Math::MatrixReal->new($N, $N);
$m->one();
return $m;
}
sub example1 {
my $m = Math::MatrixReal->new(2, 2);
$m->assign(1, 1, cplx(3, 0));
$m->assign(1, 2, cplx(2, 1));
$m->assign(2, 1, cplx(2, -1));
$m->assign(2, 2, cplx(1, 0));
return $m;
}
sub example2 {
my $m = Math::MatrixReal->new(3, 3);
$m->assign(1, 1, cplx(1, 0));
$m->assign(1, 2, cplx(1, 0));
$m->assign(1, 3, cplx(0, 0));
$m->assign(2, 1, cplx(0, 0));
$m->assign(2, 2, cplx(1, 0));
$m->assign(2, 3, cplx(1, 0));
$m->assign(3, 1, cplx(1, 0));
$m->assign(3, 2, cplx(0, 0));
$m->assign(3, 3, cplx(1, 0));
return $m;
}
sub example3 {
my $m = Math::MatrixReal->new(3, 3);
$m->assign(1, 1, cplx(0.70710677, 0));
$m->assign(1, 2, cplx(0.70710677, 0));
$m->assign(1, 3, cplx(0, 0));
$m->assign(2, 1, cplx(0, -0.70710677));
$m->assign(2, 2, cplx(0, 0.70710677));
$m->assign(2, 3, cplx(0, 0));
$m->assign(3, 1, cplx(0, 0));
$m->assign(3, 2, cplx(0, 0));
$m->assign(3, 3, cplx(0, 1));
return $m;
}</syntaxhighlight>
{{out}}
<pre>
Starting matrix:
[3, 2+i]
[2-i, 1]
Its conjugate transpose:
[3, 2+i]
[2-i, 1]
Is Hermitian? TRUE
Is normal? TRUE
Is unitary? FALSE
Starting matrix:
[1, 1, 0]
[0, 1, 1]
[1, 0, 1]
Its conjugate transpose:
[1, 0, 1]
[1, 1, 0]
[0, 1, 1]
Is Hermitian? FALSE
Is normal? TRUE
Is unitary? FALSE
Starting matrix:
[0.70710677, 0.70710677, 0]
[-0.70710677i, 0.70710677i, 0]
[0, 0, i]
Its conjugate transpose:
[0.70710677, 0.70710677i, 0]
[0.70710677, -0.70710677i, 0]
[0, 0, -i]
Is Hermitian? FALSE
Is normal? TRUE
Is unitary? TRUE
</pre>
=={{header|Phix}}==
Note this code has no testing for non-square matrices.
<!--<syntaxhighlight lang="phix">(phixonline)-->
<span style="color: #008080;">with</span> <span style="color: #008080;">javascript_semantics</span>
<span style="color: #008080;">include</span> <span style="color: #004080;">complex</span><span style="color: #0000FF;">.</span><span style="color: #000000;">e</span>
<span style="color: #008080;">procedure</span> <span style="color: #000000;">m_print</span><span style="color: #0000FF;">(</span><span style="color: #004080;">sequence</span> <span style="color: #000000;">a</span><span style="color: #0000FF;">)</span>
<span style="color: #000000;">a</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">deep_copy</span><span style="color: #0000FF;">(</span><span style="color: #000000;">a</span><span style="color: #0000FF;">)</span>
<span style="color: #004080;">integer</span> <span style="color: #000000;">l</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">length</span><span style="color: #0000FF;">(</span><span style="color: #000000;">a</span><span style="color: #0000FF;">)</span>
<span style="color: #008080;">for</span> <span style="color: #000000;">i</span><span style="color: #0000FF;">=</span><span style="color: #000000;">1</span> <span style="color: #008080;">to</span> <span style="color: #000000;">l</span> <span style="color: #008080;">do</span>
<span style="color: #008080;">for</span> <span style="color: #000000;">j</span><span style="color: #0000FF;">=</span><span style="color: #000000;">1</span> <span style="color: #008080;">to</span> <span style="color: #000000;">l</span> <span style="color: #008080;">do</span>
<span style="color: #000000;">a</span><span style="color: #0000FF;">[</span><span style="color: #000000;">i</span><span style="color: #0000FF;">][</span><span style="color: #000000;">j</span><span style="color: #0000FF;">]</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">complex_sprint</span><span style="color: #0000FF;">(</span><span style="color: #000000;">a</span><span style="color: #0000FF;">[</span><span style="color: #000000;">i</span><span style="color: #0000FF;">][</span><span style="color: #000000;">j</span><span style="color: #0000FF;">])</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">for</span>
<span style="color: #000000;">a</span><span style="color: #0000FF;">[</span><span style="color: #000000;">i</span><span style="color: #0000FF;">]</span> <span style="color: #0000FF;">=</span> <span style="color: #008000;">"["</span><span style="color: #0000FF;">&</span><span style="color: #7060A8;">join</span><span style="color: #0000FF;">(</span><span style="color: #000000;">a</span><span style="color: #0000FF;">[</span><span style="color: #000000;">i</span><span style="color: #0000FF;">],</span><span style="color: #008000;">","</span><span style="color: #0000FF;">)&</span><span style="color: #008000;">"]"</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">for</span>
<span style="color: #7060A8;">puts</span><span style="color: #0000FF;">(</span><span style="color: #000000;">1</span><span style="color: #0000FF;">,</span><span style="color: #7060A8;">join</span><span style="color: #0000FF;">(</span><span style="color: #000000;">a</span><span style="color: #0000FF;">,</span><span style="color: #008000;">"\n"</span><span style="color: #0000FF;">)&</span><span style="color: #008000;">"\n"</span><span style="color: #0000FF;">)</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">procedure</span>
<span style="color: #008080;">function</span> <span style="color: #000000;">conjugate_transpose</span><span style="color: #0000FF;">(</span><span style="color: #004080;">sequence</span> <span style="color: #000000;">a</span><span style="color: #0000FF;">)</span>
<span style="color: #004080;">sequence</span> <span style="color: #000000;">res</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">deep_copy</span><span style="color: #0000FF;">(</span><span style="color: #000000;">a</span><span style="color: #0000FF;">)</span>
<span style="color: #004080;">integer</span> <span style="color: #000000;">l</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">length</span><span style="color: #0000FF;">(</span><span style="color: #000000;">a</span><span style="color: #0000FF;">)</span>
<span style="color: #008080;">for</span> <span style="color: #000000;">i</span><span style="color: #0000FF;">=</span><span style="color: #000000;">1</span> <span style="color: #008080;">to</span> <span style="color: #000000;">l</span> <span style="color: #008080;">do</span>
<span style="color: #008080;">for</span> <span style="color: #000000;">j</span><span style="color: #0000FF;">=</span><span style="color: #000000;">1</span> <span style="color: #008080;">to</span> <span style="color: #000000;">l</span> <span style="color: #008080;">do</span>
<span style="color: #000000;">res</span><span style="color: #0000FF;">[</span><span style="color: #000000;">i</span><span style="color: #0000FF;">][</span><span style="color: #000000;">j</span><span style="color: #0000FF;">]</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">complex_conjugate</span><span style="color: #0000FF;">(</span><span style="color: #000000;">a</span><span style="color: #0000FF;">[</span><span style="color: #000000;">j</span><span style="color: #0000FF;">][</span><span style="color: #000000;">i</span><span style="color: #0000FF;">])</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">for</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">for</span>
<span style="color: #008080;">return</span> <span style="color: #000000;">res</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">function</span>
<span style="color: #008080;">function</span> <span style="color: #000000;">m_unitary</span><span style="color: #0000FF;">(</span><span style="color: #004080;">sequence</span> <span style="color: #000000;">act</span><span style="color: #0000FF;">)</span>
<span style="color: #000080;font-style:italic;">-- note: a was normal and act = a*ct already</span>
<span style="color: #004080;">integer</span> <span style="color: #000000;">l</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">length</span><span style="color: #0000FF;">(</span><span style="color: #000000;">act</span><span style="color: #0000FF;">)</span>
<span style="color: #008080;">for</span> <span style="color: #000000;">i</span><span style="color: #0000FF;">=</span><span style="color: #000000;">1</span> <span style="color: #008080;">to</span> <span style="color: #000000;">l</span> <span style="color: #008080;">do</span>
<span style="color: #008080;">for</span> <span style="color: #000000;">j</span><span style="color: #0000FF;">=</span><span style="color: #000000;">1</span> <span style="color: #008080;">to</span> <span style="color: #000000;">l</span> <span style="color: #008080;">do</span>
<span style="color: #004080;">atom</span> <span style="color: #0000FF;">{</span><span style="color: #000000;">re</span><span style="color: #0000FF;">,</span><span style="color: #000000;">im</span><span style="color: #0000FF;">}</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">act</span><span style="color: #0000FF;">[</span><span style="color: #000000;">i</span><span style="color: #0000FF;">,</span><span style="color: #000000;">j</span><span style="color: #0000FF;">]</span>
<span style="color: #000080;font-style:italic;">-- round to nearest billionth
-- (powers of 2 help the FPU out)</span>
<span style="color: #000000;">re</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">round</span><span style="color: #0000FF;">(</span><span style="color: #000000;">re</span><span style="color: #0000FF;">,</span><span style="color: #000000;">1024</span><span style="color: #0000FF;">*</span><span style="color: #000000;">1024</span><span style="color: #0000FF;">*</span><span style="color: #000000;">1024</span><span style="color: #0000FF;">)</span>
<span style="color: #000000;">im</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">round</span><span style="color: #0000FF;">(</span><span style="color: #000000;">im</span><span style="color: #0000FF;">,</span><span style="color: #000000;">1024</span><span style="color: #0000FF;">*</span><span style="color: #000000;">1024</span><span style="color: #0000FF;">*</span><span style="color: #000000;">1024</span><span style="color: #0000FF;">)</span>
<span style="color: #008080;">if</span> <span style="color: #000000;">im</span><span style="color: #0000FF;">!=</span><span style="color: #000000;">0</span>
<span style="color: #008080;">or</span> <span style="color: #0000FF;">(</span><span style="color: #000000;">i</span><span style="color: #0000FF;">=</span><span style="color: #000000;">j</span> <span style="color: #008080;">and</span> <span style="color: #000000;">re</span><span style="color: #0000FF;">!=</span><span style="color: #000000;">1</span><span style="color: #0000FF;">)</span>
<span style="color: #008080;">or</span> <span style="color: #0000FF;">(</span><span style="color: #000000;">i</span><span style="color: #0000FF;">!=</span><span style="color: #000000;">j</span> <span style="color: #008080;">and</span> <span style="color: #000000;">re</span><span style="color: #0000FF;">!=</span><span style="color: #000000;">0</span><span style="color: #0000FF;">)</span> <span style="color: #008080;">then</span>
<span style="color: #008080;">return</span> <span style="color: #000000;">0</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">if</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">for</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">for</span>
<span style="color: #008080;">return</span> <span style="color: #000000;">1</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">function</span>
<span style="color: #008080;">function</span> <span style="color: #000000;">m_mul</span><span style="color: #0000FF;">(</span><span style="color: #004080;">sequence</span> <span style="color: #000000;">a</span><span style="color: #0000FF;">,</span> <span style="color: #004080;">sequence</span> <span style="color: #000000;">b</span><span style="color: #0000FF;">)</span>
<span style="color: #004080;">sequence</span> <span style="color: #000000;">res</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">sq_mul</span><span style="color: #0000FF;">(</span><span style="color: #000000;">a</span><span style="color: #0000FF;">,</span><span style="color: #000000;">0</span><span style="color: #0000FF;">)</span>
<span style="color: #004080;">integer</span> <span style="color: #000000;">l</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">length</span><span style="color: #0000FF;">(</span><span style="color: #000000;">a</span><span style="color: #0000FF;">)</span>
<span style="color: #008080;">for</span> <span style="color: #000000;">i</span><span style="color: #0000FF;">=</span><span style="color: #000000;">1</span> <span style="color: #008080;">to</span> <span style="color: #000000;">l</span> <span style="color: #008080;">do</span>
<span style="color: #008080;">for</span> <span style="color: #000000;">j</span><span style="color: #0000FF;">=</span><span style="color: #000000;">1</span> <span style="color: #008080;">to</span> <span style="color: #000000;">l</span> <span style="color: #008080;">do</span>
<span style="color: #008080;">for</span> <span style="color: #000000;">k</span><span style="color: #0000FF;">=</span><span style="color: #000000;">1</span> <span style="color: #008080;">to</span> <span style="color: #000000;">l</span> <span style="color: #008080;">do</span>
<span style="color: #000000;">res</span><span style="color: #0000FF;">[</span><span style="color: #000000;">i</span><span style="color: #0000FF;">][</span><span style="color: #000000;">j</span><span style="color: #0000FF;">]</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">complex_add</span><span style="color: #0000FF;">(</span><span style="color: #000000;">res</span><span style="color: #0000FF;">[</span><span style="color: #000000;">i</span><span style="color: #0000FF;">][</span><span style="color: #000000;">j</span><span style="color: #0000FF;">],</span><span style="color: #7060A8;">complex_mul</span><span style="color: #0000FF;">(</span><span style="color: #000000;">a</span><span style="color: #0000FF;">[</span><span style="color: #000000;">i</span><span style="color: #0000FF;">][</span><span style="color: #000000;">k</span><span style="color: #0000FF;">],</span><span style="color: #000000;">b</span><span style="color: #0000FF;">[</span><span style="color: #000000;">k</span><span style="color: #0000FF;">][</span><span style="color: #000000;">j</span><span style="color: #0000FF;">]))</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">for</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">for</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">for</span>
<span style="color: #008080;">return</span> <span style="color: #000000;">res</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">function</span>
<span style="color: #008080;">procedure</span> <span style="color: #000000;">test</span><span style="color: #0000FF;">(</span><span style="color: #004080;">sequence</span> <span style="color: #000000;">a</span><span style="color: #0000FF;">)</span>
<span style="color: #004080;">sequence</span> <span style="color: #000000;">ct</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">conjugate_transpose</span><span style="color: #0000FF;">(</span><span style="color: #000000;">a</span><span style="color: #0000FF;">)</span>
<span style="color: #7060A8;">printf</span><span style="color: #0000FF;">(</span><span style="color: #000000;">1</span><span style="color: #0000FF;">,</span><span style="color: #008000;">"Original matrix:\n"</span><span style="color: #0000FF;">)</span>
<span style="color: #000000;">m_print</span><span style="color: #0000FF;">(</span><span style="color: #000000;">a</span><span style="color: #0000FF;">)</span>
<span style="color: #7060A8;">printf</span><span style="color: #0000FF;">(</span><span style="color: #000000;">1</span><span style="color: #0000FF;">,</span><span style="color: #008000;">"Conjugate transpose:\n"</span><span style="color: #0000FF;">)</span>
<span style="color: #000000;">m_print</span><span style="color: #0000FF;">(</span><span style="color: #000000;">ct</span><span style="color: #0000FF;">)</span>
<span style="color: #000080;font-style:italic;">-- note: rounding similar to that in m_unitary may be rqd (in a similar
-- loop in a new m_equal function) on these two equality tests,
-- but as it is, all tests pass with the builtin = operator.</span>
<span style="color: #7060A8;">printf</span><span style="color: #0000FF;">(</span><span style="color: #000000;">1</span><span style="color: #0000FF;">,</span><span style="color: #008000;">"Hermitian?: %t\n"</span><span style="color: #0000FF;">,</span><span style="color: #000000;">a</span><span style="color: #0000FF;">=</span><span style="color: #000000;">ct</span><span style="color: #0000FF;">)</span> <span style="color: #000080;font-style:italic;">-- (this one)</span>
<span style="color: #004080;">sequence</span> <span style="color: #000000;">act</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">m_mul</span><span style="color: #0000FF;">(</span><span style="color: #000000;">a</span><span style="color: #0000FF;">,</span><span style="color: #000000;">ct</span><span style="color: #0000FF;">),</span> <span style="color: #000000;">cta</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">m_mul</span><span style="color: #0000FF;">(</span><span style="color: #000000;">ct</span><span style="color: #0000FF;">,</span><span style="color: #000000;">a</span><span style="color: #0000FF;">)</span>
<span style="color: #004080;">bool</span> <span style="color: #000000;">normal</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">act</span><span style="color: #0000FF;">=</span><span style="color: #000000;">cta</span> <span style="color: #000080;font-style:italic;">-- (&this one)</span>
<span style="color: #7060A8;">printf</span><span style="color: #0000FF;">(</span><span style="color: #000000;">1</span><span style="color: #0000FF;">,</span><span style="color: #008000;">"Normal?: %t\n"</span><span style="color: #0000FF;">,</span><span style="color: #000000;">normal</span><span style="color: #0000FF;">)</span>
<span style="color: #7060A8;">printf</span><span style="color: #0000FF;">(</span><span style="color: #000000;">1</span><span style="color: #0000FF;">,</span><span style="color: #008000;">"Unitary?: %t\n\n"</span><span style="color: #0000FF;">,</span><span style="color: #000000;">normal</span> <span style="color: #008080;">and</span> <span style="color: #000000;">m_unitary</span><span style="color: #0000FF;">(</span><span style="color: #000000;">act</span><span style="color: #0000FF;">))</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">procedure</span>
<span style="color: #008080;">constant</span> <span style="color: #000000;">x</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">sqrt</span><span style="color: #0000FF;">(</span><span style="color: #000000;">2</span><span style="color: #0000FF;">)/</span><span style="color: #000000;">2</span><span style="color: #0000FF;">,</span>
<span style="color: #000000;">tests</span> <span style="color: #0000FF;">=</span> <span style="color: #0000FF;">{{{{</span><span style="color: #000000;">3</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">0</span><span style="color: #0000FF;">},{</span><span style="color: #000000;">2</span><span style="color: #0000FF;">,</span><span style="color: #000000;">1</span><span style="color: #0000FF;">}},</span>
<span style="color: #0000FF;">{{</span><span style="color: #000000;">2</span><span style="color: #0000FF;">,-</span><span style="color: #000000;">1</span><span style="color: #0000FF;">},{</span><span style="color: #000000;">1</span><span style="color: #0000FF;">,</span><span style="color: #000000;">0</span><span style="color: #0000FF;">}}},</span>
<span style="color: #0000FF;">{{{</span> <span style="color: #000000;">1</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">0</span><span style="color: #0000FF;">},{</span> <span style="color: #000000;">1</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">1</span><span style="color: #0000FF;">},{</span> <span style="color: #000000;">0</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">2</span><span style="color: #0000FF;">}},</span>
<span style="color: #0000FF;">{{</span> <span style="color: #000000;">1</span><span style="color: #0000FF;">,-</span><span style="color: #000000;">1</span><span style="color: #0000FF;">},{</span> <span style="color: #000000;">5</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">0</span><span style="color: #0000FF;">},{-</span><span style="color: #000000;">3</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">0</span><span style="color: #0000FF;">}},</span>
<span style="color: #0000FF;">{{</span> <span style="color: #000000;">0</span><span style="color: #0000FF;">,-</span><span style="color: #000000;">2</span><span style="color: #0000FF;">},{-</span><span style="color: #000000;">3</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">0</span><span style="color: #0000FF;">},{</span> <span style="color: #000000;">0</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">0</span><span style="color: #0000FF;">}}},</span>
<span style="color: #0000FF;">{{{</span><span style="color: #000000;">0.5</span><span style="color: #0000FF;">,+</span><span style="color: #000000;">0.5</span><span style="color: #0000FF;">},{</span><span style="color: #000000;">0.5</span><span style="color: #0000FF;">,-</span><span style="color: #000000;">0.5</span><span style="color: #0000FF;">}},</span>
<span style="color: #0000FF;">{{</span><span style="color: #000000;">0.5</span><span style="color: #0000FF;">,-</span><span style="color: #000000;">0.5</span><span style="color: #0000FF;">},{</span><span style="color: #000000;">0.5</span><span style="color: #0000FF;">,+</span><span style="color: #000000;">0.5</span><span style="color: #0000FF;">}}},</span>
<span style="color: #0000FF;">{{{</span> <span style="color: #000000;">1</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">0</span><span style="color: #0000FF;">},{</span> <span style="color: #000000;">1</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">0</span><span style="color: #0000FF;">},{</span> <span style="color: #000000;">0</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">0</span><span style="color: #0000FF;">}},</span>
<span style="color: #0000FF;">{{</span> <span style="color: #000000;">0</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">0</span><span style="color: #0000FF;">},{</span> <span style="color: #000000;">1</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">0</span><span style="color: #0000FF;">},{</span> <span style="color: #000000;">1</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">0</span><span style="color: #0000FF;">}},</span>
<span style="color: #0000FF;">{{</span> <span style="color: #000000;">1</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">0</span><span style="color: #0000FF;">},{</span> <span style="color: #000000;">0</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">0</span><span style="color: #0000FF;">},{</span> <span style="color: #000000;">1</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">0</span><span style="color: #0000FF;">}}},</span>
<span style="color: #0000FF;">{{{</span><span style="color: #000000;">x</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">0</span><span style="color: #0000FF;">},{</span><span style="color: #000000;">x</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">0</span><span style="color: #0000FF;">},{</span><span style="color: #000000;">0</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">0</span><span style="color: #0000FF;">}},</span>
<span style="color: #0000FF;">{{</span><span style="color: #000000;">0</span><span style="color: #0000FF;">,-</span><span style="color: #000000;">x</span><span style="color: #0000FF;">},{</span><span style="color: #000000;">0</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">x</span><span style="color: #0000FF;">},{</span><span style="color: #000000;">0</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">0</span><span style="color: #0000FF;">}},</span>
<span style="color: #0000FF;">{{</span><span style="color: #000000;">0</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">0</span><span style="color: #0000FF;">},{</span><span style="color: #000000;">0</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">0</span><span style="color: #0000FF;">},{</span><span style="color: #000000;">0</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">1</span><span style="color: #0000FF;">}}},</span>
<span style="color: #0000FF;">{{{</span><span style="color: #000000;">2</span><span style="color: #0000FF;">,</span><span style="color: #000000;">7</span><span style="color: #0000FF;">},{</span><span style="color: #000000;">9</span><span style="color: #0000FF;">,-</span><span style="color: #000000;">5</span><span style="color: #0000FF;">}},</span>
<span style="color: #0000FF;">{{</span><span style="color: #000000;">3</span><span style="color: #0000FF;">,</span><span style="color: #000000;">4</span><span style="color: #0000FF;">},{</span><span style="color: #000000;">8</span><span style="color: #0000FF;">,-</span><span style="color: #000000;">6</span><span style="color: #0000FF;">}}}}</span>
<span style="color: #7060A8;">papply</span><span style="color: #0000FF;">(</span><span style="color: #000000;">tests</span><span style="color: #0000FF;">,</span><span style="color: #000000;">test</span><span style="color: #0000FF;">)</span>
<!--</syntaxhighlight>-->
{{out}}
<pre>
Original matrix:
[3,2+i]
[2-i,1]
Conjugate transpose:
[3,2+i]
[2-i,1]
Hermitian?: true
Normal?: true
Unitary?: false
Original matrix:
[1,1+i,2i]
[1-i,5,-3]
[-2i,-3,0]
Conjugate transpose:
[1,1+i,2i]
[1-i,5,-3]
[-2i,-3,0]
Hermitian?: true
Normal?: true
Unitary?: false
Original matrix:
[0.5+0.5i,0.5-0.5i]
[0.5-0.5i,0.5+0.5i]
Conjugate transpose:
[0.5-0.5i,0.5+0.5i]
[0.5+0.5i,0.5-0.5i]
Hermitian?: false
Normal?: true
Unitary?: true
Original matrix:
[1,1,0]
[0,1,1]
[1,0,1]
Conjugate transpose:
[1,0,1]
[1,1,0]
[0,1,1]
Hermitian?: false
Normal?: true
Unitary?: false
Original matrix:
[0.707107,0.707107,0]
[-0.707107i,0.707107i,0]
[0,0,i]
Conjugate transpose:
[0.707107,0.707107i,0]
[0.707107,-0.707107i,0]
[0,0,-i]
Hermitian?: false
Normal?: true
Unitary?: true
Original matrix:
[2+7i,9-5i]
[3+4i,8-6i]
Conjugate transpose:
[2-7i,3-4i]
[9+5i,8+6i]
Hermitian?: false
Normal?: false
Unitary?: false
</pre>
=={{header|PL/I}}==
<syntaxhighlight lang="pl/i">
test: procedure options (main); /* 1 October 2012 */
declare n fixed binary;
Line 583 ⟶ 2,942:
end MMULT;
end test;
</syntaxhighlight>
Outputs from separate runs:
<pre>
Line 624 ⟶ 2,983:
Matrix is unitary
Matrix is unitary
</pre>
=={{header|PowerShell}}==
<syntaxhighlight lang="powershell">
function conjugate-transpose($a) {
$arr = @()
if($a) {
$n = $a.count - 1
if(0 -lt $n) {
$m = ($a | foreach {$_.count} | measure-object -Minimum).Minimum - 1
if( 0 -le $m) {
if (0 -lt $m) {
$arr =@(0)*($m+1)
foreach($i in 0..$m) {
$arr[$i] = foreach($j in 0..$n) {@([System.Numerics.complex]::Conjugate($a[$j][$i]))}
}
} else {$arr = foreach($row in $a) {[System.Numerics.complex]::Conjugate($row[0])}}
}
} else {$arr = foreach($row in $a) {[System.Numerics.complex]::Conjugate($row[0])}}
}
$arr
}
function multarrays-complex($a, $b) {
$c = @()
if($a -and $b) {
$n = $a.count - 1
$m = $b[0].count - 1
$c = @([System.Numerics.complex]::new(0,0))*($n+1)
foreach ($i in 0..$n) {
$c[$i] = foreach ($j in 0..$m) {
[System.Numerics.complex]$sum = [System.Numerics.complex]::new(0,0)
foreach ($k in 0..$n){$sum = [System.Numerics.complex]::Add($sum, ([System.Numerics.complex]::Multiply($a[$i][$k],$b[$k][$j])))}
$sum
}
}
}
$c
}
function identity-complex($n) {
if(0 -lt $n) {
$array = @(0) * $n
foreach ($i in 0..($n-1)) {
$array[$i] = @([System.Numerics.complex]::new(0,0)) * $n
$array[$i][$i] = [System.Numerics.complex]::new(1,0)
}
$array
} else { @() }
}
function are-eq ($a,$b) { -not (Compare-Object $a $b -SyncWindow 0)}
function show($a) {
if($a) {
0..($a.Count - 1) | foreach{ if($a[$_]){"$($a[$_])"}else{""} }
}
}
function complex($a,$b) {[System.Numerics.complex]::new($a,$b)}
$id2 = identity-complex 2
$m = @(@((complex 2 7), (complex 9 -5)),@((complex 3 4), (complex 8 -6)))
$hm = conjugate-transpose $m
$mhm = multarrays-complex $m $hm
$hmm = multarrays-complex $hm $m
"`$m ="
show $m
""
"`$hm = conjugate-transpose `$m ="
show $hm
""
"`$m * `$hm ="
show $mhm
""
"`$hm * `$m ="
show $hmm
""
"Hermitian? `$m = $(are-eq $m $hm)"
"Normal? `$m = $(are-eq $mhm $hmm)"
"Unitary? `$m = $((are-eq $id2 $hmm) -and (are-eq $id2 $mhm))"
</syntaxhighlight>
<b>Output:</b>
<pre>
$m =
(2, 7) (9, -5)
(3, 4) (8, -6)
$hm = conjugate-transpose $m =
(2, -7) (3, -4)
(9, 5) (8, 6)
$m * $hm =
(159, 0) (136, 27)
(136, -27) (125, 0)
$hm * $m =
(78, 0) (-17, -123)
(-17, 123) (206, 0)
Hermitian? $m = False
Normal? $m = False
Unitary? $m = False
</pre>
=={{header|Python}}==
Internally, matrices must be represented as rectangular tuples of tuples of complex numbers.
<syntaxhighlight lang="python">def conjugate_transpose(m):
return tuple(tuple(n.conjugate() for n in row) for row in zip(*m))
def mmul( ma, mb):
return tuple(tuple(sum( ea*eb for ea,eb in zip(a,b)) for b in zip(*mb)) for a in ma)
def mi(size):
'Complex Identity matrix'
sz = range(size)
m = [[0 + 0j for i in sz] for j in sz]
for i in range(size):
m[i][i] = 1 + 0j
return tuple(tuple(row) for row in m)
def __allsame(vector):
first, rest = vector[0], vector[1:]
return all(i == first for i in rest)
def __allnearsame(vector, eps=1e-14):
first, rest = vector[0], vector[1:]
return all(abs(first.real - i.real) < eps and abs(first.imag - i.imag) < eps
for i in rest)
def isequal(matrices, eps=1e-14):
'Check any number of matrices for equality within eps'
x = [len(m) for m in matrices]
if not __allsame(x): return False
y = [len(m[0]) for m in matrices]
if not __allsame(y): return False
for s in range(x[0]):
for t in range(y[0]):
if not __allnearsame([m[s][t] for m in matrices], eps): return False
return True
def ishermitian(m, ct):
return isequal([m, ct])
def isnormal(m, ct):
return isequal([mmul(m, ct), mmul(ct, m)])
def isunitary(m, ct):
mct, ctm = mmul(m, ct), mmul(ct, m)
mctx, mcty, cmx, ctmy = len(mct), len(mct[0]), len(ctm), len(ctm[0])
ident = mi(mctx)
return isequal([mct, ctm, ident])
def printm(comment, m):
print(comment)
fields = [['%g%+gj' % (f.real, f.imag) for f in row] for row in m]
width = max(max(len(f) for f in row) for row in fields)
lines = (', '.join('%*s' % (width, f) for f in row) for row in fields)
print('\n'.join(lines))
if __name__ == '__main__':
for matrix in [
((( 3.000+0.000j), (+2.000+1.000j)),
(( 2.000-1.000j), (+1.000+0.000j))),
((( 1.000+0.000j), (+1.000+0.000j), (+0.000+0.000j)),
(( 0.000+0.000j), (+1.000+0.000j), (+1.000+0.000j)),
(( 1.000+0.000j), (+0.000+0.000j), (+1.000+0.000j))),
((( 2**0.5/2+0.000j), (+2**0.5/2+0.000j), (+0.000+0.000j)),
(( 0.000+2**0.5/2j), (+0.000-2**0.5/2j), (+0.000+0.000j)),
(( 0.000+0.000j), (+0.000+0.000j), (+0.000+1.000j)))]:
printm('\nMatrix:', matrix)
ct = conjugate_transpose(matrix)
printm('Its conjugate transpose:', ct)
print('Hermitian? %s.' % ishermitian(matrix, ct))
print('Normal? %s.' % isnormal(matrix, ct))
print('Unitary? %s.' % isunitary(matrix, ct))</syntaxhighlight>
{{out}}
<pre>Matrix:
3+0j, 2+1j
2-1j, 1+0j
Its conjugate transpose:
3-0j, 2+1j
2-1j, 1-0j
Hermitian? True.
Normal? True.
Unitary? False.
Matrix:
1+0j, 1+0j, 0+0j
0+0j, 1+0j, 1+0j
1+0j, 0+0j, 1+0j
Its conjugate transpose:
1-0j, 0-0j, 1-0j
1-0j, 1-0j, 0-0j
0-0j, 1-0j, 1-0j
Hermitian? False.
Normal? True.
Unitary? False.
Matrix:
0.707107+0j, 0.707107+0j, 0+0j
0-0.707107j, 0+0.707107j, 0+0j
0+0j, 0+0j, 0+1j
Its conjugate transpose:
0.707107-0j, 0+0.707107j, 0-0j
0.707107-0j, 0-0.707107j, 0-0j
0-0j, 0-0j, 0-1j
Hermitian? False.
Normal? True.
Unitary? True.</pre>
=={{header|Racket}}==
<syntaxhighlight lang="racket">
#lang racket
(require math)
(define H matrix-hermitian)
(define (normal? M)
(define MH (H M))
(equal? (matrix* MH M)
(matrix* M MH)))
(define (unitary? M)
(define MH (H M))
(and (matrix-identity? (matrix* MH M))
(matrix-identity? (matrix* M MH))))
(define (hermitian? M)
(equal? (H M) M))
</syntaxhighlight>
Test:
<syntaxhighlight lang="racket">
(define M (matrix [[3.000+0.000i +2.000+1.000i]
[2.000-1.000i +1.000+0.000i]]))
(H M)
(normal? M)
(unitary? M)
(hermitian? M)
</syntaxhighlight>
Output:
<syntaxhighlight lang="racket">
(array #[#[3.0-0.0i 2.0+1.0i] #[2.0-1.0i 1.0-0.0i]])
#t
#f
#f
</syntaxhighlight>
=={{header|Raku}}==
(formerly Perl 6)
{{works with|Rakudo|2015-12-13}}
<syntaxhighlight lang="raku" line>for [ # Test Matrices
[ 1, 1+i, 2i],
[ 1-i, 5, -3],
[0-2i, -3, 0]
],
[
[1, 1, 0],
[0, 1, 1],
[1, 0, 1]
],
[
[0.707 , 0.707, 0],
[0.707i, 0-0.707i, 0],
[0 , 0, i]
]
-> @m {
say "\nMatrix:";
@m.&say-it;
my @t = @m».conj.&mat-trans;
say "\nTranspose:";
@t.&say-it;
say "Is Hermitian?\t{is-Hermitian(@m, @t)}";
say "Is Normal?\t{is-Normal(@m, @t)}";
say "Is Unitary?\t{is-Unitary(@m, @t)}";
}
sub is-Hermitian (@m, @t, --> Bool) {
so @m».Complex eqv @t».Complex
}
sub is-Normal (@m, @t, --> Bool) {
so mat-mult(@m, @t)».Complex eqv mat-mult(@t, @m)».Complex
}
sub is-Unitary (@m, @t, --> Bool) {
so mat-mult(@m, @t, 1e-3)».Complex eqv mat-ident(+@m)».Complex;
}
sub mat-trans (@m) { map { [ @m[*;$_] ] }, ^@m[0] }
sub mat-ident ($n) { [ map { [ flat 0 xx $_, 1, 0 xx $n - 1 - $_ ] }, ^$n ] }
sub mat-mult (@a, @b, \ε = 1e-15) {
my @p;
for ^@a X ^@b[0] -> ($r, $c) {
@p[$r][$c] += @a[$r][$_] * @b[$_][$c] for ^@b;
@p[$r][$c].=round(ε); # avoid floating point math errors
}
@p
}
sub say-it (@array) { $_».fmt("%9s").say for @array }</syntaxhighlight>
{{out}}
<pre>Matrix:
[ 1 1+1i 0+2i]
[ 1-1i 5 -3]
[ 0-2i -3 0]
Transpose:
[ 1 1+1i 0+2i]
[ 1-1i 5 -3]
[ 0-2i -3 0]
Is Hermitian? True
Is Normal? True
Is Unitary? False
Matrix:
[ 1 1 0]
[ 0 1 1]
[ 1 0 1]
Transpose:
[ 1 0 1]
[ 1 1 0]
[ 0 1 1]
Is Hermitian? False
Is Normal? True
Is Unitary? False
Matrix:
[ 0.707 0.707 0]
[ 0+0.707i 0-0.707i 0]
[ 0 0 0+1i]
Transpose:
[ 0.707 0-0.707i 0]
[ 0.707 0+0.707i 0]
[ 0 0 0-1i]
Is Hermitian? False
Is Normal? True
Is Unitary? True
</pre>
=={{header|REXX}}==
<syntaxhighlight lang="rexx">/*REXX program performs a conjugate transpose on a complex square matrix. */
parse arg N elements; if N==''|N=="," then N=3 /*Not specified? Then use the default.*/
k= 0; do r=1 for N
do c=1 for N; k= k+1; M.r.c= word( word(elements, k) 1, 1)
end /*c*/
end /*r*/
call showCmat 'M' ,N /*display a nicely formatted matrix. */
identity.= 0; do d=1 for N; identity.d.d= 1; end /*d*/
call conjCmat 'MH', "M" ,N /*conjugate the M matrix ───► MH */
call showCmat 'MH' ,N /*display a nicely formatted matrix. */
say 'M is Hermitian: ' word('no yes', isHermitian('M', "MH", N) + 1)
call multCmat 'M', 'MH', 'MMH', N /*multiple the two matrices together. */
call multCmat 'MH', 'M', 'MHM', N /* " " " " " */
say ' M is Normal: ' word('no yes', isHermitian('MMH', "MHM", N) + 1)
say ' M is Unary: ' word('no yes', isUnary('M', N) + 1)
say 'MMH is Unary: ' word('no yes', isUnary('MMH', N) + 1)
say 'MHM is Unary: ' word('no yes', isUnary('MHM', N) + 1)
exit /*stick a fork in it, we're all done. */
/*──────────────────────────────────────────────────────────────────────────────────────*/
cP: procedure; arg ',' c; return word( strip( translate(c, , 'IJ') ) 0, 1)
rP: procedure; parse arg r ','; return word( r 0, 1) /*◄──maybe return a 0 ↑ */
/*──────────────────────────────────────────────────────────────────────────────────────*/
conjCmat: parse arg matX,matY,rows 1 cols; call normCmat matY, rows
do r=1 for rows; _=
do c=1 for cols; v= value(matY'.'r"."c)
rP= rP(v); cP= -cP(v); call value matX'.'c"."r, rP','cP
end /*c*/
end /*r*/; return
/*──────────────────────────────────────────────────────────────────────────────────────*/
isHermitian: parse arg matX,matY,rows 1 cols; call normCmat matX, rows
call normCmat matY, rows
do r=1 for rows; _=
do c=1 for cols
if value(matX'.'r"."c) \= value(matY'.'r"."c) then return 0
end /*c*/
end /*r*/; return 1
/*──────────────────────────────────────────────────────────────────────────────────────*/
isUnary: parse arg matX,rows 1 cols
do r=1 for rows; _=
do c=1 for cols; z= value(matX'.'r"."c); rP= rP(z); cP= cP(z)
if abs( sqrt( rP(z) **2 + cP(z)**2) - (r==c)) >= .0001 then return 0
end /*c*/
end /*r*/; return 1
/*──────────────────────────────────────────────────────────────────────────────────────*/
multCmat: parse arg matA,matB,matT,rows 1 cols; call value matT'.', 0
do r=1 for rows; _=
do c=1 for cols
do k=1 for cols; T= value(matT'.'r"."c); Tr= rP(T); Tc= cP(T)
A= value(matA'.'r"."k); Ar= rP(A); Ac= cP(A)
B= value(matB'.'k"."c); Br= rP(B); Bc= cP(B)
Pr= Ar*Br - Ac*Bc; Pc= Ac*Br + Ar*Bc; Tr= Tr+Pr; Tc= Tc+Pc
call value matT'.'r"."c,Tr','Tc
end /*k*/
end /*c*/
end /*r*/; return
/*──────────────────────────────────────────────────────────────────────────────────────*/
normCmat: parse arg matN,rows 1 cols
do r=1 to rows; _=
do c=1 to cols; v= translate( value(matN'.'r"."c), , "IiJj")
parse upper var v real ',' cplx
if real\=='' then real= real / 1
if cplx\=='' then cplx= cplx / 1; if cplx=0 then cplx=
if cplx\=='' then cplx= cplx"j"
call value matN'.'r"."c, strip(real','cplx, "T", ',')
end /*c*/
end /*r*/; return
/*──────────────────────────────────────────────────────────────────────────────────────*/
showCmat: parse arg matX,rows,cols; if cols=='' then cols= rows; @@= left('', 6)
say; say center('matrix' matX, 79, '─'); call normCmat matX, rows, cols
do r=1 to rows; _=
do c=1 to cols; _= _ @@ left( value(matX'.'r"."c), 9)
end /*c*/
say _
end /*r*/; say; return
/*──────────────────────────────────────────────────────────────────────────────────────*/
sqrt: procedure; parse arg x; if x=0 then return 0; d=digits(); numeric form; h=d+6
numeric digits; parse value format(x,2,1,,0) 'E0' with g 'E' _ .; g=g *.5'e'_ % 2
m.=9; do j=0 while h>9; m.j=h; h=h%2+1; end /*j*/
do k=j+5 to 0 by -1; numeric digits m.k; g=(g+x/g)*.5; end /*k*/; return g</syntaxhighlight>
{{out|output|text= when using the default input:}}
<pre>
───────────────────────────────────matrix M────────────────────────────────────
1 1 1
1 1 1
1 1 1
───────────────────────────────────matrix MH───────────────────────────────────
1 1 1
1 1 1
1 1 1
M is Hermitian: yes
M is Normal: yes
M is Unary: no
MMH is Unary: no
MHM is Unary: no
</pre>
{{out|output|text= when using the input of: <tt> 3 .7071 .7071 0 0,.7071 0,-.7071 0 0 0 0,1 </tt>}}
<pre>
───────────────────────────────────matrix M────────────────────────────────────
0.7071 0.7071 0
0,0.7071j 0,-0.7071 0
0 0 0,1j
───────────────────────────────────matrix MH───────────────────────────────────
0.7071 0,-0.7071 0
0.7071 0,0.7071j 0
0 0 0,-1j
M is Hermitian: no
M is Normal: yes
M is Unary: no
MMH is Unary: yes
MHM is Unary: yes
</pre>
=={{header|RPL}}==
Although basic, RPL's matrix handling capabilities help to keep the code compact but still readable.
{{works with|Halcyon Calc|4.2.7}}
≪ - ABS 1E-10 < ≫
´SAME?´ STO
≪ DUP TRN → m mh
≪ m mh SAME? "Hermitian. " "" IFTE
m mh * mh m * SAME? "Normal. " "" IFTE +
m INV mh SAME? "Unitary. " "" IFTE +
≫
≫
´CNJTRN’ STO
[[(3,0) (2,1)][(2,-1) (1,0)]]
'Hm' STO
[[1 1 0][0 1 1][1 0 1]]
'Nm' STO
≪ [[1 1 0][(0,1) (0,-1) 0][0 0 0]] 2 √ 2 / * {3 3} (0,1) PUT ≫
'Um' STO
Hm CNJTRN
Nm CNJTRN
Um CNJTRN
{{out}}
<pre>
3: "Hermitian. Normal. "
2: "Normal. "
1: "Normal. Unitary. "
</pre>
=={{header|Ruby}}==
{{works with|Ruby|2.0}}
<syntaxhighlight lang="ruby">require 'matrix'
# Start with some matrix.
Line 653 ⟶ 3,505:
print ' normal? false'
print ' unitary? false'
end</
Note: Ruby 1.9 had a bug in the Matrix#hermitian? method. It's fixed in 2.0.
=={{header|Rust}}==
Uses external crate 'num', version 0.1.34
<syntaxhighlight lang="rust">
extern crate num; // crate for complex numbers
use num::complex::Complex;
use std::ops::Mul;
use std::fmt;
#[derive(Debug, PartialEq)]
struct Matrix<f32> {
grid: [[Complex<f32>; 2]; 2], // used to represent matrix
}
impl Matrix<f32> { // implements a method call for calculating the conjugate transpose
fn conjugate_transpose(&self) -> Matrix<f32> {
Matrix {grid: [[self.grid[0][0].conj(), self.grid[1][0].conj()],
[self.grid[0][1].conj(), self.grid[1][1].conj()]]}
}
}
impl Mul for Matrix<f32> { // implements '*' (multiplication) for the matrix
type Output = Matrix<f32>;
fn mul(self, other: Matrix<f32>) -> Matrix<f32> {
Matrix {grid: [[self.grid[0][0]*other.grid[0][0] + self.grid[0][1]*other.grid[1][0],
self.grid[0][0]*other.grid[0][1] + self.grid[0][1]*other.grid[1][1]],
[self.grid[1][0]*other.grid[0][0] + self.grid[1][1]*other.grid[1][0],
self.grid[1][0]*other.grid[1][0] + self.grid[1][1]*other.grid[1][1]]]}
}
}
impl Copy for Matrix<f32> {} // implemented to prevent 'moved value' errors in if statements below
impl Clone for Matrix<f32> {
fn clone(&self) -> Matrix<f32> {
*self
}
}
impl fmt::Display for Matrix<f32> { // implemented to make output nicer
fn fmt(&self, f: &mut fmt::Formatter) -> fmt::Result {
write!(f, "({}, {})\n({}, {})", self.grid[0][0], self.grid[0][1], self.grid[1][0], self.grid[1][1])
}
}
fn main() {
let a = Matrix {grid: [[Complex::new(3.0, 0.0), Complex::new(2.0, 1.0)],
[Complex::new(2.0, -1.0), Complex::new(1.0, 0.0)]]};
let b = Matrix {grid: [[Complex::new(0.5, 0.5), Complex::new(0.5, -0.5)],
[Complex::new(0.5, -0.5), Complex::new(0.5, 0.5)]]};
test_type(a);
test_type(b);
}
fn test_type(mat: Matrix<f32>) {
let identity = Matrix {grid: [[Complex::new(1.0, 0.0), Complex::new(0.0, 0.0)],
[Complex::new(0.0, 0.0), Complex::new(1.0, 0.0)]]};
let mat_conj = mat.conjugate_transpose();
println!("Matrix: \n{}\nConjugate transpose: \n{}", mat, mat_conj);
if mat == mat_conj {
println!("Hermitian?: TRUE");
} else {
println!("Hermitian?: FALSE");
}
if mat*mat_conj == mat_conj*mat {
println!("Normal?: TRUE");
} else {
println!("Normal?: FALSE");
}
if mat*mat_conj == identity {
println!("Unitary?: TRUE");
} else {
println!("Unitary?: FALSE");
}
}</syntaxhighlight>
Output:
<pre>
Matrix:
(3+0i, 2+1i)
(2-1i, 1+0i)
Conjugate transpose:
(3+0i, 2+1i)
(2-1i, 1+0i)
Hermitian?: TRUE
Normal?: TRUE
Unitary?: FALSE
Matrix:
(0.5+0.5i, 0.5-0.5i)
(0.5-0.5i, 0.5+0.5i)
Conjugate transpose:
(0.5-0.5i, 0.5+0.5i)
(0.5+0.5i, 0.5-0.5i)
Hermitian?: FALSE
Normal?: TRUE
Unitary?: TRUE
</pre>
=={{header|Scala}}==
<syntaxhighlight lang="scala">object ConjugateTranspose {
case class Complex(re: Double, im: Double) {
def conjugate(): Complex = Complex(re, -im)
def +(other: Complex) = Complex(re + other.re, im + other.im)
def *(other: Complex) = Complex(re * other.re - im * other.im, re * other.im + im * other.re)
override def toString(): String = {
if (im < 0) {
s"${re}${im}i"
} else {
s"${re}+${im}i"
}
}
}
case class Matrix(val entries: Vector[Vector[Complex]]) {
def *(other: Matrix): Matrix = {
new Matrix(
Vector.tabulate(entries.size, other.entries(0).size)((r, c) => {
val rightRow = entries(r)
val leftCol = other.entries.map(_(c))
rightRow.zip(leftCol)
.map{ case (x, y) => x * y } // multiply pair-wise
.foldLeft(new Complex(0,0)){ case (x, y) => x + y } // sum over all
})
)
}
def conjugateTranspose(): Matrix = {
new Matrix(
Vector.tabulate(entries(0).size, entries.size)((r, c) => entries(c)(r).conjugate)
)
}
def isHermitian(): Boolean = {
this == conjugateTranspose()
}
def isNormal(): Boolean = {
val ct = conjugateTranspose()
this * ct == ct * this
}
def isIdentity(): Boolean = {
val entriesWithIndexes = for (r <- 0 until entries.size; c <- 0 until entries(r).size) yield (r, c, entries(r)(c))
entriesWithIndexes.forall { case (r, c, x) =>
if (r == c) {
x == Complex(1.0, 0.0)
} else {
x == Complex(0.0, 0.0)
}
}
}
def isUnitary(): Boolean = {
(this * conjugateTranspose()).isIdentity()
}
override def toString(): String = {
entries.map(" " + _.mkString("[", ",", "]")).mkString("[\n", "\n", "\n]")
}
}
def main(args: Array[String]): Unit = {
val m = new Matrix(
Vector.fill(3, 3)(new Complex(Math.random() * 2 - 1.0, Math.random() * 2 - 1.0))
)
println("Matrix: " + m)
println("Conjugate Transpose: " + m.conjugateTranspose())
println("Hermitian: " + m.isHermitian())
println("Normal: " + m.isNormal())
println("Unitary: " + m.isUnitary())
}
}</syntaxhighlight>
{{out}}
<pre>
Matrix: [
[-0.7679977131543951-0.439979346567841i,-0.6011221529373452+0.510336881376179i,-0.22458301626795674-0.2036390034398219i]
[-0.29309032295973036+0.3034337168992096i,-0.06392399629070344-0.8178102917845342i,0.06006452944412022-0.6141208421036348i]
[0.34841978725201117+0.3778314407778909i,0.6768867572228499+0.7323625144544055i,-0.8246879334889017-0.009443253424316733i]
]
Conjugate Transpose: [
[-0.7679977131543951+0.439979346567841i,-0.29309032295973036-0.3034337168992096i,0.34841978725201117-0.3778314407778909i]
[-0.6011221529373452-0.510336881376179i,-0.06392399629070344+0.8178102917845342i,0.6768867572228499-0.7323625144544055i]
[-0.22458301626795674+0.2036390034398219i,0.06006452944412022+0.6141208421036348i,-0.8246879334889017+0.009443253424316733i]
]
Hermitian: false
Normal: false
Unitary: false
</pre>
=={{header|Sidef}}==
{{trans|Raku}}
<syntaxhighlight lang="ruby">func is_Hermitian (Array m, Array t) -> Bool { m == t }
func mat_mult (Array a, Array b, Number ε = -3) {
var p = []
for r, c in (^a ~X ^b[0]) {
for k in (^b) {
p[r][c] := 0 += (a[r][k] * b[k][c]) -> round!(ε)
}
}
return p
}
func mat_trans (Array m) {
var r = []
for i,j in (^m ~X ^m[0]) {
r[j][i] = m[i][j]
}
return r
}
func mat_ident (Number n) {
^n -> map {|i|
[i.of(0)..., 1, (n - i - 1).of(0)...]
}
}
func is_Normal (Array m, Array t) -> Bool {
mat_mult(m, t) == mat_mult(t, m)
}
func is_Unitary (Array m, Array t) -> Bool {
mat_mult(m, t) == mat_ident(m.len)
}
func say_it (Array a) {
a.each {|b|
b.map { "%9s" % _ }.join(' ').say
}
}
[
[
[ 1, 1+1i, 2i],
[1-1i, 5, -3],
[0-2i, -3, 0]
],
[
[1, 1, 0],
[0, 1, 1],
[1, 0, 1]
],
[
[0.707 , 0.707, 0],
[0.707i, -0.707i, 0],
[0 , 0, 1i]
]
].each { |m|
say "\nMatrix:"
say_it(m)
var t = mat_trans(m.map{.map{.conj}})
say "\nTranspose:"
say_it(t)
say "Is Hermitian?\t#{is_Hermitian(m, t)}"
say "Is Normal?\t#{is_Normal(m, t)}"
say "Is Unitary?\t#{is_Unitary(m, t)}"
}</syntaxhighlight>
{{out}}
<pre>
Matrix:
1 1+i 2i
1-i 5 -3
-2i -3 0
Transpose:
1 1+i 2i
1-i 5 -3
-2i -3 0
Is Hermitian? true
Is Normal? true
Is Unitary? false
Matrix:
1 1 0
0 1 1
1 0 1
Transpose:
1 0 1
1 1 0
0 1 1
Is Hermitian? false
Is Normal? true
Is Unitary? false
Matrix:
0.707 0.707 0
0.707i -0.707i 0
0 0 i
Transpose:
0.707 -0.707i 0
0.707 0.707i 0
0 0 -i
Is Hermitian? false
Is Normal? true
Is Unitary? true
</pre>
=={{header|Sparkling}}==
Sparkling has support for basic complex algebraic operations, but complex matrix operations are not in the standard library.
<syntaxhighlight lang="sparkling"># Computes conjugate transpose of M
let conjTransp = function conjTransp(M) {
return map(range(sizeof M[0]), function(row) {
return map(range(sizeof M), function(col) {
return cplx_conj(M[col][row]);
});
});
};
# Helper for cplxMatMul
let cplxVecScalarMul = function cplxVecScalarMul(A, B, row, col) {
var M = { "re": 0.0, "im": 0.0 };
let N = sizeof A;
for (var i = 0; i < N; i++) {
let P = cplx_mul(A[row][i], B[i][col]);
M = cplx_add(M, P);
}
return M;
};
# Multiplies matrices A and B
# A and B are assumed to be square and of the same size,
# this condition is not checked.
let cplxMatMul = function cplxMatMul(A, B) {
var R = {};
let N = sizeof A;
for (var row = 0; row < N; row++) {
R[row] = {};
for (var col = 0; col < N; col++) {
R[row][col] = cplxVecScalarMul(A, B, row, col);
}
}
return R;
};
# Helper for creating an array representing a complex number
# given its textual representation
let _ = function makeComplex(str) {
let sep = indexof(str, "+", 1);
if sep < 0 {
sep = indexof(str, "-", 1);
}
let reStr = substrto(str, sep);
let imStr = substrfrom(str, sep);
return { "re": tofloat(reStr), "im": tofloat(imStr) };
};
# Formats a complex matrix
let printCplxMat = function printCplxMat(M) {
foreach(M, function(i, row) {
foreach(row, function(j, elem) {
printf(" %.2f%+.2fi", elem.re, elem.im);
});
print();
});
};
# A Hermitian matrix
let H = {
{ _("3+0i"), _("2+1i") },
{ _("2-1i"), _("0+0i") }
};
# A normal matrix
let N = {
{ _("1+0i"), _("1+0i"), _("0+0i") },
{ _("0+0i"), _("1+0i"), _("1+0i") },
{ _("1+0i"), _("0+0i"), _("1+0i") }
};
# A unitary matrix
let U = {
{ _("0.70710678118+0i"), _("0.70710678118+0i"), _("0+0i") },
{ _("0-0.70710678118i"), _("0+0.70710678118i"), _("0+0i") },
{ _("0+0i"), _("0+0i"), _("0+1i") }
};
print("Hermitian matrix:\nH = ");
printCplxMat(H);
print("H* = ");
printCplxMat(conjTransp(H));
print();
print("Normal matrix:\nN = ");
printCplxMat(N);
print("N* = ");
printCplxMat(conjTransp(N));
print("N* x N = ");
printCplxMat(cplxMatMul(conjTransp(N), N));
print("N x N* = ");
printCplxMat(cplxMatMul(N, conjTransp(N)));
print();
print("Unitary matrix:\nU = ");
printCplxMat(U);
print("U* = ");
printCplxMat(conjTransp(U));
print("U x U* = ");
printCplxMat(cplxMatMul(U, conjTransp(U)));
print();</syntaxhighlight>
=={{header|Stata}}==
In Mata, the ' operator is always the conjugate transpose. To get only the matrix transpose without complex conjugate, use the [ transposeonly] function.
<syntaxhighlight lang="stata">
: a=1,2i\3i,4
: a
1 2
+-----------+
1 | 1 2i |
2 | 3i 4 |
+-----------+
: a'
1 2
+-------------+
1 | 1 -3i |
2 | -2i 4 |
+-------------+
: transposeonly(a)
1 2
+-----------+
1 | 1 3i |
2 | 2i 4 |
+-----------+
: a*a'==a'*a
0
: a'==a
0
: a'*a==I(rows(a))
0
</syntaxhighlight>
=={{header|Tcl}}==
Line 659 ⟶ 3,964:
{{tcllib|math::complexnumbers}}
{{tcllib|struct::matrix}}
<
package require math::complexnumbers
Line 715 ⟶ 4,020:
}
return $mat
}</
Using these tools to test for the properties described in the task:
<
if {[$matrix rows] != [$matrix columns]} {
# Must be square!
Line 768 ⟶ 4,073:
$mmh destroy
return $result
}</
<!-- Wot, no demonstration? -->
=={{header|Wren}}==
{{libheader|Wren-fmt}}
{{libheader|Wren-complex}}
Although the third example is in fact a unitary matrix, the ''isUnitary'' method of the above module returns false.
This is because the methods in the module work as accurately as they can within the confines of 64-bit floating point arithmetic and don't therefore allow for the small rounding error that occurs due to the use of the irrational number, sqrt(2).
However, if we use the ''almostEquals'' method with the default tolerance of 1.0e-14, then we do get a ''true'' result.
<syntaxhighlight lang="wren">import "./complex" for Complex, CMatrix
import "./fmt" for Fmt
var cm1 = CMatrix.new(
[
[Complex.new(3), Complex.new(2, 1)],
[Complex.new(2, -1), Complex.one ]
]
)
var cm2 = CMatrix.fromReals([ [1, 1, 0], [0, 1, 1], [1, 0, 1] ])
var x = 2.sqrt/2
var cm3 = CMatrix.new(
[
[Complex.new(x), Complex.new(x), Complex.zero],
[Complex.new(0, -x), Complex.new(0, x), Complex.zero],
[Complex.zero, Complex.zero, Complex.imagOne]
]
)
for (cm in [cm1, cm2, cm3]) {
System.print("Matrix:")
Fmt.mprint(cm, 5, 3)
System.print("\nConjugate transpose:")
Fmt.mprint(cm.conjTranspose, 5, 3)
System.print("\nHermitian : %(cm.isHermitian)")
System.print("Normal : %(cm.isNormal)")
System.print("Unitary : %(cm.isUnitary)")
System.print()
}
System.print("For the final example if we use a tolerance of 1e-14:")
var cm4 = cm3 * cm3.conjTranspose
var id = CMatrix.identity(3)
System.print("Unitary : %(cm4.almostEquals(id))")</syntaxhighlight>
{{out}}
<pre>
Matrix:
|3.000 + 0.000i 2.000 + 1.000i|
|2.000 - 1.000i 1.000 + 0.000i|
Conjugate transpose:
|3.000 + 0.000i 2.000 + 1.000i|
|2.000 - 1.000i 1.000 + 0.000i|
Hermitian : true
Normal : true
Unitary : false
Matrix:
|1.000 + 0.000i 1.000 + 0.000i 0.000 + 0.000i|
|0.000 + 0.000i 1.000 + 0.000i 1.000 + 0.000i|
|1.000 + 0.000i 0.000 + 0.000i 1.000 + 0.000i|
Conjugate transpose:
|1.000 + 0.000i 0.000 + 0.000i 1.000 + 0.000i|
|1.000 + 0.000i 1.000 + 0.000i 0.000 + 0.000i|
|0.000 + 0.000i 1.000 + 0.000i 1.000 + 0.000i|
Hermitian : false
Normal : true
Unitary : false
Matrix:
|0.707 + 0.000i 0.707 + 0.000i 0.000 + 0.000i|
|0.000 - 0.707i 0.000 + 0.707i 0.000 + 0.000i|
|0.000 + 0.000i 0.000 + 0.000i 0.000 + 1.000i|
Conjugate transpose:
|0.707 + 0.000i 0.000 + 0.707i 0.000 + 0.000i|
|0.707 + 0.000i 0.000 - 0.707i 0.000 + 0.000i|
|0.000 + 0.000i 0.000 + 0.000i 0.000 - 1.000i|
Hermitian : false
Normal : true
Unitary : false
For the final example if we use a tolerance of 1e-14:
Unitary : true
</pre>
| |||