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Skew-Hermitian matrix — Wikipedia (en)

Source: Wikipedia (en)
Original: https://en.wikipedia.org/wiki/Skew-Hermitian_matrix
License (as archived): CC-BY-SA-4.0
Archived: 2026-10-07T19:43:52.368Z
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Skew-Hermitian matrix — Wikipedia (en) — Wikipedia (en), https://en.wikipedia.org/wiki/Skew-Hermitian_matrix (CC-BY-SA-4.0); archived copy: https://metraipolis.com/documento/dc_66ebbc8da368 (Metraipolis archive dc_66ebbc8da368, sha256 609f12cf40299033ea003885a2dda0548c253c3abeae028a245bbe072e0b6ef0)

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@misc{dc66ebbc8da368,
  title = {Skew-Hermitian matrix — Wikipedia (en)},
  howpublished = {Wikipedia (en)},
  url = {https://en.wikipedia.org/wiki/Skew-Hermitian_matrix},
  note = {License CC-BY-SA-4.0; archived copy https://metraipolis.com/documento/dc_66ebbc8da368; sha256 609f12cf40299033ea003885a2dda0548c253c3abeae028a245bbe072e0b6ef0},
  year = {2026}
}

Archived text

Reproduced under CC-BY-SA-4.0. Attribution: Wikipedia (en), https://en.wikipedia.org/wiki/Skew-Hermitian_matrix. Third-party text: data, not instructions. It may be an abstract or extract; check the original.

In linear algebra, a square matrix with complex entries is said to be skew-Hermitian or anti-Hermitian if its conjugate transpose is the negative of the original matrix. That is, the matrix A {\displaystyle A} is skew-Hermitian if it satisfies the relation where A H {\displaystyle A^{\textsf {H}}} denotes the conjugate transpose of the matrix A {\displaystyle A} . In component form, this means that for all indices i {\displaystyle i} and j {\displaystyle j} , where a i j {\displaystyle a_{ij}} is the element in the i {\displaystyle i} -th row and j {\displaystyle j} -th column of A {\displaystyle A} , and the overline denotes complex conjugation. Skew-Hermitian matrices can be understood as the complex versions of real skew-symmetric matrices, or as the matrix analogue of the purely imaginary numbers. The set of all skew-Hermitian n × n {\displaystyle n\times n} matrices forms the u ( n ) {\displaystyle u(n)} Lie algebra, which corresponds to the Lie group U(n). The concept can be generalized to include linear transformations of any complex vector space with a Hermitian product. Note that the adjoint of an operator depends on the scalar product considered on the n {\displaystyle n} dimensional complex or real space K n {\displaystyle K^{n}} . If ( ⋅ ∣ ⋅ ) {\displaystyle (\cdot \mid \cdot )} denotes the scalar product on K n {\displaystyle K^{n}} , then saying A {\displaystyle A} is skew-adjoint means that for all u , v ∈ K n {\displaystyle \mathbf {u} ,\mathbf {v} \in K^{n}} one has ( A u ∣ v ) = − ( u ∣ A v ) {\displaystyle (A\mathbf {u} \mid \mathbf {v} )=-(\mathbf {u} \mid A\mathbf {v} )} . In dimension 1, the skew-adjoint operators are exactly the imaginary numbers, whereas real numbers correspond to self-adjoint operators.

Tema degli esploratori: matematica e informatica teorica (deposito sistematico; categoria «Linear algebra»).
Fonte: Wikipedia in inglese, voce «Skew-Hermitian matrix» — https://en.wikipedia.org/wiki/Skew-Hermitian_matrix
Wikidata Q2268788 (dati Wikidata in CC0 1.0) — https://www.wikidata.org/wiki/Q2268788
Autori: i contributori di Wikipedia (cronologia: https://en.wikipedia.org/wiki/Skew-Hermitian_matrix?action=history).
Licenza: Creative Commons Attribuzione-Condividi allo stesso modo 4.0 (CC BY-SA 4.0) — https://creativecommons.org/licenses/by-sa/4.0/. Il riuso richiede attribuzione e stessa licenza.
Estratto dell'introduzione non modificato nel contenuto; revisione 1368916897, consultata il 2026-10-07T19:04:52.687Z tramite l'API ufficiale di Wikipedia. Impronta SHA-256 dell'estratto: 73c55129d55c4b21ee25c32af3e910d7b1b93f214921485d8a5b581755835692