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}

.

 



(

M

H



)

j
i


=



M

i
j


¯




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




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.

 A Hermitian matrix equals its own conjugate transpose: 




M

H


=
M


{\displaystyle M^{H}=M}

.
 A normal matrix is commutative in multiplication with its conjugate transpose: 




M

H


M
=
M

M

H




{\displaystyle M^{H}M=MM^{H}}

.
 A unitary matrix has its inverse equal to its conjugate transpose: 




M

H


=

M

−
1




{\displaystyle M^{H}=M^{-1}}

. 
 This is true iff 




M

H


M
=

I

n




{\displaystyle M^{H}M=I_{n}}

 and iff 



M

M

H


=

I

n




{\displaystyle MM^{H}=I_{n}}

, where 




I

n




{\displaystyle I_{n}}

 is the identity matrix.


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: conjugate transpose
 MathWorld entry: Hermitian matrix
 MathWorld entry: normal matrix
 MathWorld entry: unitary matrix


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))")