ConjugateTranspose

ConjugateTranspose[m]

or gives the conjugate transpose of .

Details and Options

ConjugateTranspose[m] is equivalent to Conjugate[Transpose[m]]. »
can be entered as ct or \[ConjugateTranspose].
ConjugateTranspose[m] can also be given as , where can be entered as hc or \[HermitianConjugate].
ConjugateTranspose[m,spec] gives Conjugate[Transpose[m,spec]].

Examples

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Basic Examples (2)

Conjugate transpose of a complex-valued matrix:

Enter using ct:

Scope (2)

Conjugate transpose a sparse array:

The conjugate transpose is sparse:

ConjugateTranspose works for symbolic matrices:

ComplexExpand assumes all variables are real:

Generalizations & Extensions (1)

ConjugateTranspose works similarly to Transpose for tensors:

Conjugate and transpose the first two dimensions:

Conjugate and transpose the first and third dimensions:

Applications (1)

is a random complex matrix:

Find the QRDecomposition of :

is unitary, so its inverse is

Reconstruct from the decomposition:

Properties & Relations (2)

ConjugateTranspose[m] is equivalent to Conjugate[Transpose[m]]:

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

is the matrix product of and :

so is Hermitian:

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