Problem Statement

Suppose that a matrix

M

{\displaystyle M}

contains complex numbers. Then the conjugate transpose of

M

{\displaystyle M}

is a matrix

M

H

{\displaystyle M^{H}}

containing the complex conjugates of the matrix transposition of

M

{\displaystyle M}

.

This means that row

j

{\displaystyle j}

, column

i

{\displaystyle i}

of the conjugate transpose equals the complex conjugate of row

i

{\displaystyle i}

, column

j

{\displaystyle j}

of the original matrix.

In the next list,

M

{\displaystyle M}

must also be a square matrix.

Given some matrix of complex numbers, find its conjugate transpose. Also determine if the matrix is a:

Let's start with the solution:

USING: kernel math.functions math.matrices sequences ;
IN: rosetta.hermitian

: conj-t ( matrix -- conjugate-transpose )
    flip [ [ conjugate ] map ] map ;

: hermitian-matrix? ( matrix -- ? )
    dup conj-t = ;

: normal-matrix? ( matrix -- ? )
    dup conj-t [ m. ] [ swap m. ] 2bi = ;

: unitary-matrix? ( matrix -- ? )
    [ dup conj-t m. ] [ length identity-matrix ] bi = ;