({{, (RefLink[ConjugateTranspose, paclet:ref/ConjugateTranspose])[m]
or (m^†) gives the conjugate transpose of m. }})

OpenersNotesSection(Spacer1)Details and Options

(RefLink[ConjugateTranspose, paclet:ref/ConjugateTranspose])[m] is equivalent to (RefLink[Conjugate, paclet:ref/Conjugate])[(RefLink[Transpose, paclet:ref/Transpose])[m]]. »

† can be entered as (Key0[esc]) ct (Key0[esc]) or \[ConjugateTranspose].

(RefLink[ConjugateTranspose, paclet:ref/ConjugateTranspose])[m] can also be given as (m^†), where † can be entered as (Key0[esc]) hc (Key0[esc]) or \[HermitianConjugate].

(RefLink[ConjugateTranspose, paclet:ref/ConjugateTranspose])[m,spec] gives (RefLink[Conjugate, paclet:ref/Conjugate])[(RefLink[Transpose, paclet:ref/Transpose])[m,spec]].

OpenersPrimaryExamplesSection(Spacer1)Examples  (8)

OpenersExampleSection0(Spacer1)Basic Examples  (2)

Conjugate transpose of a complex-valued matrix:

m={{1,2 I,3}, {3+4 I,5,I}};

m//MatrixForm

ConjugateTranspose[m]//MatrixForm

Enter using (Key0[esc]) ct (Key0[esc]):

{{1,2+I},{3,4}}\[ConjugateTranspose]

OpenersExampleSection1(Spacer1)Scope  (2)

Conjugate transpose a sparse array:

s = SparseArray[{{i_, i_}->1, {i_, j_}/;Abs[i-j]==1->i + j I}, {5,5}]

MatrixForm[s]

The conjugate transpose is sparse:

st = ConjugateTranspose[s]

MatrixForm[st]

(RefLink[ConjugateTranspose, paclet:ref/ConjugateTranspose]) works for symbolic matrices:

ct = ConjugateTranspose[Array[(Subscript[a, ##]) + (Subscript[b, ##])I&, {2,2}]]

(RefLink[ComplexExpand, paclet:ref/ComplexExpand]) assumes all variables are real:

ComplexExpand[ct]

OpenersExampleSection2(Spacer1)Generalizations & Extensions  (1)

(RefLink[ConjugateTranspose, paclet:ref/ConjugateTranspose]) works similarly to (RefLink[Transpose, paclet:ref/Transpose]) for tensors:

Block[{j = 0}, t = Table[(++j) + I(++j), {2},{3},{4}]]

Conjugate and transpose the first two dimensions:

ConjugateTranspose[t]

Conjugate and transpose the first and third dimensions:

ConjugateTranspose[t, {3,2,1}]

OpenersExampleSection3(Spacer1)Applications  (1)

m is a random complex matrix:

m = RandomComplex[1 + I, {4,4}];

Find the (RefLink[QRDecomposition, paclet:ref/QRDecomposition]) of m:

{q, r} = QRDecomposition[m];

q is unitary, so its inverse is (q^†)

Chop[q.(q^\[ConjugateTranspose])]

Reconstruct m from the decomposition:

(q^\[ConjugateTranspose]).r-m//Chop

OpenersExampleSection4(Spacer1)Properties & Relations  (2)

(RefLink[ConjugateTranspose, paclet:ref/ConjugateTranspose])[m] is equivalent to (RefLink[Conjugate, paclet:ref/Conjugate])[(RefLink[Transpose, paclet:ref/Transpose])[m]]:

m = RandomComplex[1 + I, {2,3}];

ConjugateTranspose[m] - Conjugate[Transpose[m]]

The product of a matrix and its conjugate transpose is Hermitian:

m = RandomComplex[1 + I, {3,4}];

h is the matrix product of m and (m^†):

h = m.(m^\[ConjugateTranspose]);

(h^†)==(((m.(m^†)))^†)==((((m^†)))^†).(m^†)==m.(m^†)==h so h is Hermitian:

h == (h^\[ConjugateTranspose])

({{DocumentationFooterIconsRelatedFunction.png, ({{See Also}, {(RefLink[Conjugate, paclet:ref/Conjugate])  ▪  (RefLink[Transpose, paclet:ref/Transpose])  ▪  (RefLink[Inverse, paclet:ref/Inverse])  ▪  (RefLink[HermitianMatrixQ, paclet:ref/HermitianMatrixQ])}})}})

({{DocumentationFooterIconsRelatedTutorial.png, ({{Tutorials}, {▪(RefLinkPlain[Basic Matrix Operations, paclet:tutorial/BasicMatrixOperations])}})}})

({{DocumentationFooterIconsRelatedGuide.png, ({{Related Guides}, {▪(RefLinkPlain[Matrix Operations, paclet:guide/MatrixOperations])}, {▪(RefLinkPlain[Complex Numbers, paclet:guide/ComplexNumbers])}})}})