Conjugate transpose: Difference between revisions

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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

Ada

<lang Ada>with Ada.Text_IO; use Ada.Text_IO; with Ada.Complex_Text_IO; use Ada.Complex_Text_IO; with Ada.Numerics.Complex_Types; use Ada.Numerics.Complex_Types; with Ada.Numerics.Complex_Arrays; use Ada.Numerics.Complex_Arrays; procedure ConTrans is

  subtype CM is Complex_Matrix;
  S2O2 : constant Float := 0.7071067811865;

  procedure Print (mat : CM) is begin
     for row in mat'Range(1) loop for col in mat'Range(2) loop
        Put(mat(row,col), Exp=>0, Aft=>4);
     end loop; New_Line; end loop;
  end Print;

  function almostzero(mat : CM; tol : Float) return Boolean is begin
     for row in mat'Range(1) loop for col in mat'Range(2) loop
        if abs(mat(row,col)) > tol then return False; end if;
     end loop; end loop;
     return True;
  end almostzero;

  procedure Examine (mat : CM) is
     CT : CM := Conjugate (Transpose(mat));
     isherm, isnorm, isunit : Boolean;
  begin
     isherm := almostzero(mat-CT, 1.0e-6);
     isnorm := almostzero(mat*CT-CT*mat, 1.0e-6);
     isunit := almostzero(CT-Inverse(mat), 1.0e-6);
     Print(mat);
     Put_Line("Conjugate transpose:"); Print(CT);
     Put_Line("Hermitian?: " & isherm'Img);
     Put_Line("Normal?: " & isnorm'Img);
     Put_Line("Unitary?: " & isunit'Img);
  end Examine;

  hmat : CM := ((3.0+0.0*i, 2.0+1.0*i), (2.0-1.0*i, 1.0+0.0*i));
  nmat : CM := ((1.0+0.0*i, 1.0+0.0*i, 0.0+0.0*i),
                (0.0+0.0*i, 1.0+0.0*i, 1.0+0.0*i),
                (1.0+0.0*i, 0.0+0.0*i, 1.0+0.0*i));
  umat : CM := ((S2O2+0.0*i, S2O2+0.0*i, 0.0+0.0*i),
                (0.0+S2O2*i, 0.0-S2O2*i, 0.0+0.0*i),
                (0.0+0.0*i, 0.0+0.0*i, 0.0+1.0*i));

begin

  Put_Line("hmat:"); Examine(hmat); New_Line;
  Put_Line("nmat:"); Examine(nmat); New_Line;
  Put_Line("umat:"); Examine(umat);

end ConTrans;</lang>

Output:

hmat:
( 3.0000, 0.0000)( 2.0000, 1.0000)
( 2.0000,-1.0000)( 1.0000, 0.0000)
Conjugate transpose:
( 3.0000,-0.0000)( 2.0000, 1.0000)
( 2.0000,-1.0000)( 1.0000,-0.0000)
Hermitian?: TRUE
Normal?: TRUE
Unitary?: FALSE

nmat:
( 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 transpose:
( 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

umat:
( 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 transpose:
( 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

C

<lang c>/*28th August, 2012 Abhishek Ghosh

Uses C99 specified complex.h, complex datatype has to be defined and operation provided if used on non-C99 compilers */

include<stdlib.h>
include<stdio.h>
include<complex.h>

typedef struct {

 int rows, cols;
 complex **z;

} matrix;

matrix transpose (matrix a) {

 int i, j;
 matrix b;

 b.rows = a.cols;
 b.cols = a.rows;

 b.z = malloc (b.rows * sizeof (complex *));

 for (i = 0; i < b.rows; i++)
   {
     b.z[i] = malloc (b.cols * sizeof (complex));
     for (j = 0; j < b.cols; j++)
       {
         b.z[i][j] = conj (a.z[j][i]);
       }
   }

 return b;

}

int isHermitian (matrix a) {

 int i, j;
 matrix b = transpose (a);

 if (b.rows == a.rows && b.cols == a.cols)
   {
     for (i = 0; i < b.rows; i++)
       {
         for (j = 0; j < b.cols; j++)
           {
             if (b.z[i][j] != a.z[i][j])
               return 0;
           }
       }
   }

 else
   return 0;

 return 1;

}

matrix multiply (matrix a, matrix b) {

 matrix c;
 int i, j;

 if (a.cols == b.rows)
   {
     c.rows = a.rows;
     c.cols = b.cols;

     c.z = malloc (c.rows * (sizeof (complex *)));

     for (i = 0; i < c.rows; i++)
       {
         c.z[i] = malloc (c.cols * sizeof (complex));
         c.z[i][j] = 0 + 0 * I;
         for (j = 0; j < b.cols; j++)
           {
             c.z[i][j] += a.z[i][j] * b.z[j][i];
           }
       }

 return c;

}

int isNormal (matrix a) {

 int i, j;
 matrix a_ah, ah_a;

 if (a.rows != a.cols)
   return 0;

 a_ah = multiply (a, transpose (a));
 ah_a = multiply (transpose (a), a);

 for (i = 0; i < a.rows; i++)
   {
     for (j = 0; j < a.cols; j++)
       {
         if (a_ah.z[i][j] != ah_a.z[i][j])
           return 0;
       }
   }

 return 1;

}

int isUnitary (matrix a) {

 matrix b;
 int i, j;
 if (isNormal (a) == 1)
   {
     b = multiply (a, transpose(a));

     for (i = 0; i < b.rows; i++)
       {
         for (j = 0; j < b.cols; j++)
           {
             if ((i == j && b.z[i][j] != 1) || (i != j && b.z[i][j] != 0))
               return 0;
           }
       }
     return 1;
   }
 return 0;

}

int main () {

 complex z = 3 + 4 * I;
 matrix a, aT;
 int i, j;
 printf ("Enter rows and columns :");
 scanf ("%d%d", &a.rows, &a.cols);

 a.z = malloc (a.rows * sizeof (complex *));
 printf ("Randomly Generated Complex Matrix A is : ");
 for (i = 0; i < a.rows; i++)
   {
     printf ("\n");
     a.z[i] = malloc (a.cols * sizeof (complex));
     for (j = 0; j < a.cols; j++)
       {
         a.z[i][j] = rand () % 10 + rand () % 10 * I;
         printf ("\t%f + %fi", creal (a.z[i][j]), cimag (a.z[i][j]));
       }
   }

 aT = transpose (a);

 printf ("\n\nTranspose of Complex Matrix A is : ");
 for (i = 0; i < aT.rows; i++)
   {
     printf ("\n");
     aT.z[i] = malloc (aT.cols * sizeof (complex));
     for (j = 0; j < aT.cols; j++)
       {
         aT.z[i][j] = rand () % 10 + rand () % 10 * I;
         printf ("\t%f + %fi", creal (aT.z[i][j]), cimag (aT.z[i][j]));
       }
   }

 printf ("\n\nComplex Matrix A %s hermitian",
         isHermitian (a) == 1 ? "is" : "is not");
 printf ("\n\nComplex Matrix A %s unitary",
         isUnitary (a) == 1 ? "is" : "is not");
 printf ("\n\nComplex Matrix A %s normal",
         isNormal (a) == 1 ? "is" : "is not");

 return 0;

}</lang>

Output:

Enter rows and columns :3 3
Randomly Generated Complex Matrix A is :
        3.000000 + 6.000000i    7.000000 + 5.000000i    3.000000 + 5.000000i
        6.000000 + 2.000000i    9.000000 + 1.000000i    2.000000 + 7.000000i
        0.000000 + 9.000000i    3.000000 + 6.000000i    0.000000 + 6.000000i

Transpose of Complex Matrix A is :
        2.000000 + 6.000000i    1.000000 + 8.000000i    7.000000 + 9.000000i
        2.000000 + 0.000000i    2.000000 + 3.000000i    7.000000 + 5.000000i
        9.000000 + 2.000000i    2.000000 + 8.000000i    9.000000 + 7.000000i

Complex Matrix A is not hermitian

Complex Matrix A is not unitary

Complex Matrix A is not normal

Common Lisp

<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))) )

</lang>

Output:

(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

D

Translation of: Python

A well typed and mostly imperative version:

<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));
   }

}</lang>

Output:

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.

Alternative Version

A more functional version that contains some typing problems (same output). <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));
   }

}</lang>

Factor

Before the fix to Factor bug #484, m. gave the wrong answer and this code failed. Factor 0.94 is too old to work.

Works with: Factor version development (future 0.95)

<lang factor>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 = ;</lang>

Usage:

USE: rosetta.hermitian
IN: scratchpad { { C{ 1 2 } 0 }
                 { 0 C{ 3 4 } } }
               [ hermitian-matrix? . ]
               [ normal-matrix? . ]
               [ unitary-matrix? . ] tri
f
t
f

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 :

gfortran -std=f2008 -Wall -fopenmp -ffree-form -fall-intrinsics -fimplicit-none f.f08 -o f

<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 </lang>

-*- 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

Go

<lang go>package main

import (

   "fmt"
   "math"
   "math/cmplx"

)

// a type to represent matrices type matrix struct {

   ele  []complex128
   cols int

}

// conjugate transpose, implemented here as a method on the matrix type. func (m *matrix) conjTranspose() *matrix {

   r := &matrix{make([]complex128, len(m.ele)), len(m.ele) / m.cols}
   rx := 0
   for _, e := range m.ele {
       r.ele[rx] = cmplx.Conj(e)
       rx += r.cols
       if rx >= len(r.ele) {
           rx -= len(r.ele) - 1
       }
   }
   return r

}

// program to demonstrate capabilites on example matricies func main() {

   show("h", matrixFromRows([][]complex128{
       {3, 2 + 1i},
       {2 - 1i, 1}}))

   show("n", matrixFromRows([][]complex128{
       {1, 1, 0},
       {0, 1, 1},
       {1, 0, 1}}))

   show("u", matrixFromRows([][]complex128{
       {math.Sqrt2 / 2, math.Sqrt2 / 2, 0},
       {math.Sqrt2 / -2i, math.Sqrt2 / 2i, 0},
       {0, 0, 1i}}))

}

func show(name string, m *matrix) {

   m.print(name)
   ct := m.conjTranspose()
   ct.print(name + "_ct")

   fmt.Println("Hermitian:", m.equal(ct, 1e-14))

   mct := m.mult(ct)
   ctm := ct.mult(m)
   fmt.Println("Normal:", mct.equal(ctm, 1e-14))

   i := eye(m.cols)
   fmt.Println("Unitary:", mct.equal(i, 1e-14) && ctm.equal(i, 1e-14))

}

// two constructors func matrixFromRows(rows [][]complex128) *matrix {

   m := &matrix{make([]complex128, len(rows)*len(rows[0])), len(rows[0])}
   for rx, row := range rows {
       copy(m.ele[rx*m.cols:(rx+1)*m.cols], row)
   }
   return m

}

func eye(n int) *matrix {

   r := &matrix{make([]complex128, n*n), n}
   n++
   for x := 0; x < len(r.ele); x += n {
       r.ele[x] = 1
   }
   return r

}

// print method outputs matrix to stdout func (m *matrix) print(heading string) {

   fmt.Print("\n", heading, "\n")
   for e := 0; e < len(m.ele); e += m.cols {
       fmt.Printf("%6.3f ", m.ele[e:e+m.cols])
       fmt.Println()
   }

}

// equal method uses ε to allow for floating point error. func (a *matrix) equal(b *matrix, ε float64) bool {

   for x, aEle := range a.ele {
       if math.Abs(real(aEle)-real(b.ele[x])) > math.Abs(real(aEle))*ε ||
           math.Abs(imag(aEle)-imag(b.ele[x])) > math.Abs(imag(aEle))*ε {
           return false
       }
   }
   return true

}

// mult method taken from matrix multiply task func (m1 *matrix) mult(m2 *matrix) (m3 *matrix) {

   m3 = &matrix{make([]complex128, (len(m1.ele)/m1.cols)*m2.cols), m2.cols}
   for m1c0, m3x := 0, 0; m1c0 < len(m1.ele); m1c0 += m1.cols {
       for m2r0 := 0; m2r0 < m2.cols; m2r0++ {
           for m1x, m2x := m1c0, m2r0; m2x < len(m2.ele); m2x += m2.cols {
               m3.ele[m3x] += m1.ele[m1x] * m2.ele[m2x]
               m1x++
           }
           m3x++
       }
   }
   return m3

}</lang> Output:

h
[( 3.000+0.000i) (+2.000+1.000i)] 
[( 2.000-1.000i) (+1.000+0.000i)] 

h_ct
[( 3.000-0.000i) (+2.000+1.000i)] 
[( 2.000-1.000i) (+1.000-0.000i)] 
Hermitian: true
Normal: true
Unitary: false

n
[( 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)] 

n_ct
[( 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

u
[( 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)] 

u_ct
[( 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

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)

  hermitian=: -: ct
  normal =: (X~ -: X) ct
  unitary=: ct -: %.

  (hermitian,normal,unitary)&.>HERMITIAN;NORMAL;UNITARY

+-----+-----+-----+ |1 1 0|0 1 0|0 1 1| +-----+-----+-----+</lang>

jq

Works with: jq version 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: <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 ;</lang>

(2) Operations on real/complex numbers <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;</lang>

(3) Array operations <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]) ));</lang>

(4) Matrix operations <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;</lang>

Conjugate transposition

<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)  ; </lang>

Examples

<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</lang>

Output:

<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</lang>

Julia

Julia has a built-in matrix type, and the conjugate-transpose of a complex matrix A is simply: <lang julia>A'</lang> (similar to Matlab). You can check whether A is Hermitian via the built-in function <lang julia>ishermitian(A)</lang> 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 <lang julia>isnormal(A) = size(A,1) == size(A,2) && A'*A == A*A' isunitary(A) = size(A,1) == size(A,2) && A'*A == eye(A)</lang>

Maple

The commands HermitianTranspose and IsUnitary are provided by the LinearAlgebra package. <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);</lang> Output:

                               [  3    2 + I]
                               [            ]
                               [2 - I    1  ]

                               [  3    2 + I]
                               [            ]
                               [2 - I    1  ]

                                    true

                                    true

                                    false

Mathematica / Wolfram Language

<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)

m = {{1, 2I, 3}, {3+4I, 5, I}}; m //MatrixForm -> (1 2I 3 3+4I 5 I)

ConjugateTranspose[m] //MatrixForm -> (1 3-4I -2I 5 3 -I)

{HermitianMatrixQ@#, NormalMatrixQ@#, UnitaryQ@#}&@m -> {False, False, False}</lang>

PARI/GP

<lang>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</lang>

Perl 6

Works with: Rakudo version 2015-12-13

<lang perl6>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 }</lang>

Output:

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

PL/I

<lang PL/I> test: procedure options (main); /* 1 October 2012 */

  declare n fixed binary;

  put ('Conjugate a complex square matrix.');
  put skip list ('What is the order of the matrix?:');
  get (n);
  begin;
     declare (M, MH, MM, MM_MMH, MM_MHM, IDENTITY)(n,n) fixed complex;
     declare i fixed binary;

     IDENTITY = 0; do i = 1 to n; IDENTITY(I,I) = 1; end;
     put skip list ('Please type the matrix:');
     get list (M);
     do i = 1 to n;
        put skip list (M(i,*));
     end;
     do i = 1 to n;
        MH(i,*) = conjg(M(*,i));
     end;
     put skip list ('The conjugate transpose is:');
     do i = 1 to n;
        put skip list (MH(i,*));
     end;
     if all(M=MH) then
        put skip list ('Matrix is Hermitian');
     call MMULT(M, MH, MM_MMH);
     call MMULT(MH, M, MM_MHM);

     if all(MM_MMH = MM_MHM) then
        put skip list ('Matrix is Normal');

     if all(ABS(MM_MMH - IDENTITY) < 0.0001) then
        put skip list ('Matrix is unitary');
     if all(ABS(MM_MHM - IDENTITY) < 0.0001) then
        put skip list ('Matrix is unitary');
  end;

MMULT: procedure (M, MH, MM);

  declare (M, MH, MM)(*,*) fixed complex;
  declare (i, j, n) fixed binary;

  n = hbound(M,1);
  do i = 1 to n;
     do j = 1 to n;
        MM(i,j) = sum(M(i,*) * MH(*,j) );
     end;
  end;

end MMULT; end test; </lang> Outputs from separate runs:

Please type the matrix: 

       1+0I                    1+0I                    1+0I       
       1+0I                    1+0I                    1+0I       
       1+0I                    1+0I                    1+0I       
The conjugate transpose is: 
       1-0I                    1-0I                    1-0I       
       1-0I                    1-0I                    1-0I       
       1-0I                    1-0I                    1-0I       
Matrix is Hermitian 
Matrix is Normal 

       1+0I                    1+0I                    0+0I
       0+0I                    1+0I                    1+0I       
       1+0I                    0+0I                    1+0I       
The conjugate transpose is: 
       1-0I                    0-0I                    1-0I       
       1-0I                    1-0I                    0-0I       
       0-0I                    1-0I                    1-0I       
Matrix is Normal

Next test performed with declaration of matrixes changed to decimal precision (10,5).

Please type the matrix:

      0.70710+0.00000I        0.70710+0.00000I        0.00000+0.00000I
      0.00000+0.70710I        0.00000-0.70710I        0.00000+0.00000I
      0.00000+0.00000I        0.00000+0.00000I        0.00000+1.00000I
    
The conjugate transpose is: 
      0.70710-0.00000I        0.00000-0.70710I        0.00000-0.00000I
      0.70710-0.00000I        0.00000+0.70710I        0.00000-0.00000I
      0.00000-0.00000I        0.00000-0.00000I        0.00000-1.00000I

Matrix is Normal 
Matrix is unitary 
Matrix is unitary

PowerShell

<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))" </lang> Output:

$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

Python

Internally, matrices must be represented as rectangular tuples of tuples of complex numbers. <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))</lang>

Output:

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.

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))

</lang> Test: <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) </lang> Output: <lang racket> (array #[#[3.0-0.0i 2.0+1.0i] #[2.0-1.0i 1.0-0.0i]])

t
f
f

</lang>

REXX

<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</lang>

output when using the default input:

───────────────────────────────────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

output when using the input of: 3 .7071 .7071 0 0,.7071 0,-.7071 0 0 0 0,1

───────────────────────────────────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

Ruby

Works with: Ruby version 2.0

<lang ruby>require 'matrix'

Start with some matrix.

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

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

Find the conjugate transpose.
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> Note: Ruby 1.9 had a bug in the Matrix#hermitian? method. It's fixed in 2.0.

Rust

Uses external crate 'num', version 0.1.34 <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");
   }

}</lang> Output:

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

Scala

<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())
 }

}</lang>

Output:

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

Sparkling

Sparkling has support for basic complex algebraic operations, but complex matrix operations are not in the standard library.

<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

Multiplies matrices A and B
A and B are assumed to be square and of the same size,
this condition is not checked.

Helper for creating an array representing a complex number
given its textual representation

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();</lang>

Tcl

Tcl's matrixes (in Tcllib) do not assume that the contents are numeric at all. As such, they do not provide mathematical operations over them and this considerably increases the complexity of the code below. Note the use of lambda terms to simplify access to the complex number package.

Library: Tcllib (Package: math::complexnumbers)

Library: Tcllib (Package: struct::matrix)

<lang tcl>package require struct::matrix package require math::complexnumbers

proc complexMatrix.equal {m1 m2 {epsilon 1e-14}} {

   if {[$m1 rows] != [$m2 rows] || [$m1 columns] != [$m2 columns]} {

return 0

   }
   # Compute the magnitude of the difference between two complex numbers
   set ceq [list apply {{epsilon a b} {

expr {[mod [- $a $b]] < $epsilon}

   } ::math::complexnumbers} $epsilon]
   for {set i 0} {$i<[$m1 columns]} {incr i} {

for {set j 0} {$j<[$m1 rows]} {incr j} { if {![{*}$ceq [$m1 get cell $i $j] [$m2 get cell $i $j]]} { return 0 } }

   }
   return 1

}

proc complexMatrix.multiply {a b} {

   if {[$a columns] != [$b rows]} {
       error "incompatible sizes"
   }
   # Simplest to use a lambda in the complex NS
   set cpm {{sum a b} {

+ $sum [* $a $b]

   } ::math::complexnumbers}
   set c0 [math::complexnumbers::complex 0.0 0.0];   # Complex zero
   set c [struct::matrix]
   $c add columns [$b columns]
   $c add rows [$a rows]
   for {set i 0} {$i < [$a rows]} {incr i} {
       for {set j 0} {$j < [$b columns]} {incr j} {
           set sum $c0

foreach rv [$a get row $i] cv [$b get column $j] { set sum [apply $cpm $sum $rv $cv]

$c set cell $j $i $sum

       }
   }
   return $c

}

proc complexMatrix.conjugateTranspose {matrix} {

   set mat [struct::matrix]
   $mat = $matrix
   $mat transpose
   for {set c 0} {$c < [$mat columns]} {incr c} {

for {set r 0} {$r < [$mat rows]} {incr r} { set val [$mat get cell $c $r] $mat set cell $c $r [math::complexnumbers::conj $val] }

   }
   return $mat

}</lang> Using these tools to test for the properties described in the task: <lang tcl>proc isHermitian {matrix {epsilon 1e-14}} {

   if {[$matrix rows] != [$matrix columns]} {

# Must be square! return 0

   }
   set cc [complexMatrix.conjugateTranspose $matrix]
   set result [complexMatrix.equal $matrix $cc $epsilon]
   $cc destroy
   return $result

}

proc isNormal {matrix {epsilon 1e-14}} {

   if {[$matrix rows] != [$matrix columns]} {

# Must be square! return 0

   }
   set mh [complexMatrix.conjugateTranspose $matrix]
   set mhm [complexMatrix.multiply $mh $matrix]
   set mmh [complexMatrix.multiply $matrix $mh]
   $mh destroy
   set result [complexMatrix.equal $mhm $mmh $epsilon]
   $mhm destroy
   $mmh destroy
   return $result

}

proc isUnitary {matrix {epsilon 1e-14}} {

   if {[$matrix rows] != [$matrix columns]} {

# Must be square! return 0

   }
   set mh [complexMatrix.conjugateTranspose $matrix]
   set mhm [complexMatrix.multiply $mh $matrix]
   set mmh [complexMatrix.multiply $matrix $mh]
   $mh destroy
   set result [complexMatrix.equal $mhm $mmh $epsilon]
   $mhm destroy
   if {$result} {

set id [struct::matrix] $id = $matrix; # Just for its dimensions for {set c 0} {$c < [$id columns]} {incr c} { for {set r 0} {$r < [$id rows]} {incr r} { $id set cell $c $r \ [math::complexnumbers::complex [expr {$c==$r}] 0] } } set result [complexMatrix.equal $mmh $id $epsilon] $id destroy

   }
   $mmh destroy
   return $result

}</lang>