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Antihermitian[{1,2}]

represents the symmetry of an antihermitian matrix.

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Antihermitian

Antihermitian[{1,2}]

represents the symmetry of an antihermitian matrix.

Details

  • An antihermitian matrix is also known as a skew-Hermitian matrix.
  • A square matrix m is antihermitian if ConjugateTranspose[m]-m.

Examples

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Basic Examples  (2)

This matrix is antihermitian:

Wolfram Language code: TensorSymmetry[{{3I, 1 + 2I}, {-1 + 2I, 5I}}]

Find conditions for which a matrix is antihermitian:

Wolfram Language code: {{0, b}, {I - 3, -5I}}∈Matrices[{2, 2}, Complexes, Antihermitian[{1, 2}]]

Scope  (2)

Use Antihermitian[…] as a symmetry for matrix domains:

Wolfram Language code: Matrices[{d, d}, Complexes, Antihermitian[{1, 2}]]

Use the specification to simplify symbolic matrix expressions:

Wolfram Language code: TensorExpand[ConjugateTranspose[m], Assumptions -> m∈%]

Symmetrize matrices with respect to antihermitian symmetry:

Wolfram Language code: m = {{9 I, -3 + 2 I}, {6 - 10 I, 4 + 7 I}};
Wolfram Language code: Symmetrize[m, Antihermitian[{1, 2}]]
Wolfram Language code: Normal[%]
Wolfram Language code: AntihermitianMatrixQ[%]

Applications  (1)

Take a 3×3 matrix of complexes:

Wolfram Language code: m = {{-3 + I, 1 + 5 I, 9 - 7 I}, {2 - 10 I, 5 - 4 I, -8 + 7 I}, {-1 + 3 I, 10 I, 3 - 5 I}};

It is not an antihermitian matrix:

Wolfram Language code: AntihermitianMatrixQ[m]

Compute its antihermitian part:

Wolfram Language code: s = Symmetrize[m, Antihermitian[{1, 2}]]
Wolfram Language code: Normal[s]
Wolfram Language code: AntihermitianMatrixQ[%]

Properties & Relations  (2)

Antihermitian[slots] for an array of real entries automatically converts into Antisymmetric[slots]:

Wolfram Language code: Matrices[{3, 3}, Reals, Antihermitian[{1, 2}]]

The diagonal elements of an antihermitian matrix are pure imaginary:

Wolfram Language code: Symmetrize[{{1 + I, 2 + 2I}, {3 + 3I, 4 + 4I}}, Antihermitian[{1, 2}]]
Wolfram Language code: I Diagonal[%]∈Reals
Wolfram Research (2020), Antihermitian, Wolfram Language function, https://reference.wolfram.com/language/ref/Antihermitian.html.

Text

Wolfram Research (2020), Antihermitian, Wolfram Language function, https://reference.wolfram.com/language/ref/Antihermitian.html.

CMS

Wolfram Language. 2020. "Antihermitian." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/Antihermitian.html.

APA

Wolfram Language. (2020). Antihermitian. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/Antihermitian.html

BibTeX

@misc{reference.wolfram_2026_antihermitian, author="Wolfram Research", title="{Antihermitian}", year="2020", howpublished="\url{https://reference.wolfram.com/language/ref/Antihermitian.html}", note=[Accessed: 12-June-2026]}

BibLaTeX

@online{reference.wolfram_2026_antihermitian, organization={Wolfram Research}, title={Antihermitian}, year={2020}, url={https://reference.wolfram.com/language/ref/Antihermitian.html}, note=[Accessed: 12-June-2026]}