Conjugate transpose

Conjugate transpose is a draft programming task. It is not yet considered ready to be promoted as a complete task, for reasons that should be found in its talk page.

Suppose that a matrix $M$ contains complex numbers. Then the conjugate transpose of $M$ is a matrix $M^{H}$ containing the complex conjugates of the matrix transposition of $M$.

$(M^{H})_{ji}={\overline {M_{ij}}}$

This means that row $j$, column $i$ of the conjugate transpose equals the complex conjugate of row $i$, column $j$ of the original matrix.

In the next list, $M$ must also be a square matrix.

• A Hermitian matrix equals its own conjugate transpose: $M^{H}=M$.
• A normal matrix is commutative in multiplication with its conjugate transpose: $M^{H}M=MM^{H}$.
• A unitary matrix has its inverse equal to its conjugate transpose: $M^{H}=M^{-1}$. This is true iff $M^{H}M=I_{n}$ and iff $MM^{H}=I_{n}$, where $I_{n}$ is the identity matrix.

Given some matrix of complex numbers, find its conjugate transpose. Also determine if it is a Hermitian matrix, normal matrix, or a unitary matrix.

• MathWorld: conjugate transpose, Hermitian matrix, normal matrix, unitary matrix

J

Solution: <lang j> ct =: +@|: NB. Conjugate transpose (ct A is A_ct)</lang> Examples: <lang j> X =: +/ . * NB. Matrix Multiply (x)

  HERMITIAN =:  3 2j1 ,: 2j_1 1  
  (-: ct) HERMITIAN               NB.  A_ct = A

1

  NORMAL    =:  1 1 0 , 0 1 1 ,: 1 0 1
  ((X~ -: X) ct) NORMAL           NB. A_ct x A = A x A_ct

1

  UNITARY   =:  (-:%:2) * 1 1 0 , 0j_1 0j1 0 ,: 0 0 0j1 * %:2
  (ct -: %.)  UNITARY             NB.  A_ct = A^-1

1</lang>

Reference (example matrices for other langs to use):<lang j> HERMITIAN;NORMAL;UNITARY +--------+-----+--------------------------+ | 3 2j1|1 1 0| 0.707107 0.707107 0| |2j_1 1|0 1 1|0j_0.707107 0j0.707107 0| | |1 0 1| 0 0 0j1| +--------+-----+--------------------------+

  NB. In J, PjQ is P + Q*i and the 0.7071... is sqrt(2)</lang>

Ruby

This example is incorrect. Please fix the code and remove this message. Details: Matrix#hermitian? in MRI uses a different definition of Hermitian matrix: it only checks $M_{ji}={\overline {M_{ij}}}$ for $i\neq j$.

Works with: Ruby version 1.9.3

<lang ruby>require 'matrix'

1. Start with some matrix.

i = Complex::I matrix = Matrix[[i, 0, 0],

               [0, i, 0],
               [0, 0, i]]

1. Find the conjugate transpose.
2. Matrix#conjugate appeared in Ruby 1.9.2.

conjt = matrix.conj.t # aliases for matrix.conjugate.tranpose print 'conjugate tranpose: '; puts conjt

if matrix.square?

 # These predicates appeared in Ruby 1.9.3.
 print 'Hermitian? '; puts matrix.hermitian?
 print '   normal? '; puts matrix.normal?
 print '  unitary? '; puts matrix.unitary?

else

 # Matrix is not square. These predicates would
 # raise ExceptionForMatrix::ErrDimensionMismatch.
 print 'Hermitian? false'
 print '   normal? false'
 print '  unitary? false'

end</lang>