Conjugate transpose
Suppose that a matrix contains complex numbers. Then the conjugate transpose of is a matrix containing the complex conjugates of the matrix transposition of .
This means that row , column of the conjugate transpose equals the complex conjugate of row , column of the original matrix.
In the next list, must also be a square matrix.
- A Hermitian matrix equals its own conjugate transpose: .
- A normal matrix is commutative in multiplication with its conjugate transpose: .
- A unitary matrix has its inverse equal to its conjugate transpose: . This is true iff and iff , where 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
Factor
<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>
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)</lang>
Mathematica
<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>
Ruby
<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>
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.
<lang tcl>package require struct::matrix package require math::complexnumbers
proc 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 $mat
}
proc complexMatrix.equal {m1 m2 {epsilon 1e-14}} {
if {[$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 $j]]} { return 0 } }
} return 1
}
proc isHermitian {matrix {epsilon 1e-14}} {
if {[$matrix rows] != [$matrix columns]} {
# Must be square! return 0
} set cc [conjugateTranspose $matrix] set result [complexMatrix.equal $matrix $cc $epsilon] $cc destroy return $result
}
proc complexMatrix.multiply {a b} {
if {[$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 $c
}
proc isNormal {matrix {epsilon 1e-14}} {
if {[$matrix rows] != [$matrix columns]} {
# Must be square! return 0
} set mh [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 [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>