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Hermitian[{1,2}]

represents the symmetry of a Hermitian matrix.

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Hermitian

Hermitian[{1,2}]

represents the symmetry of a Hermitian matrix.

Details

  • A Hermitian matrix is also known as a self-adjoint matrix.
  • A square matrix m is Hermitian if ConjugateTranspose[m]m.

Examples

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Basic Examples  (2)

This matrix is Hermitian:

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

Find conditions for which a matrix is Hermitian:

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

Scope  (2)

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

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

Use the specification to simplify symbolic matrix expressions:

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

Symmetrize matrices with respect to Hermitian symmetry:

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

Applications  (2)

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 a Hermitian matrix:

Wolfram Language code: HermitianMatrixQ[m]

Compute its Hermitian part:

Wolfram Language code: h = Symmetrize[m, Hermitian[{1, 2}]]
Wolfram Language code: Normal[h]
Wolfram Language code: HermitianMatrixQ[%]

Find the Hermitian matrix with minimum 2-norm (largest singular value) such that the matrix is positive semidefinite:

Wolfram Language code: ConvexOptimization[Norm[x], VectorGreaterEqual[{x, {{1, I}, {-I, 1}}}, "SemidefiniteCone"], Element[x, Matrices[{2, 2}, Complexes, Hermitian[{1, 2}]]]]

Properties & Relations  (2)

Hermitian[slots] for an array of real entries automatically converts into Symmetric[slots]:

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

The diagonal elements of a Hermitian matrix are real:

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

Text

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

CMS

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

APA

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

BibTeX

@misc{reference.wolfram_2026_hermitian, author="Wolfram Research", title="{Hermitian}", year="2020", howpublished="\url{https://reference.wolfram.com/language/ref/Hermitian.html}", note=[Accessed: 13-June-2026]}

BibLaTeX

@online{reference.wolfram_2026_hermitian, organization={Wolfram Research}, title={Hermitian}, year={2020}, url={https://reference.wolfram.com/language/ref/Hermitian.html}, note=[Accessed: 13-June-2026]}