Editing Hermitian matrix (section)

{{Short description|Matrix equal to its conjugate-transpose
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{{For|matrices with symmetry over the [[real number]] field|Symmetric matrix}}
{{Use American English|date=January 2019}}

In [[mathematics]], a '''Hermitian matrix''' (or '''self-adjoint matrix''') is a [[complex number|complex]] [[square matrix]] that is equal to its own [[conjugate transpose]]—that is, the element in the {{mvar|i}}-th row and {{mvar|j}}-th column is equal to the [[complex conjugate]] of the element in the {{mvar|j}}-th row and {{mvar|i}}-th column, for all indices {{mvar|i}} and {{mvar|j}}:
<math display =block>A \text{ is Hermitian} \quad \iff \quad a_{ij} = \overline{a_{ji}}</math>

or in matrix form:
<math display=block>A \text{ is Hermitian} \quad \iff \quad A = \overline {A^\mathsf{T}}.</math>

Hermitian matrices can be understood as the complex extension of real [[symmetric matrix|symmetric matrices]].

If the [[conjugate transpose]] of a matrix <math>A</math> is denoted by <math>A^\mathsf{H},</math> then the Hermitian property can be written concisely as

<math display=block>A \text{ is Hermitian} \quad \iff \quad A = A^\mathsf{H}</math>

Hermitian matrices are named after [[Charles Hermite]],<ref>{{Citation |last=Archibald |first=Tom |title=VI.47 Charles Hermite |date=2010-12-31 |url=https://www.degruyter.com/document/doi/10.1515/9781400830398.773a/html |work=The Princeton Companion to Mathematics |pages=773 |editor-last=Gowers |editor-first=Timothy |access-date=2023-11-15 |publisher=Princeton University Press |doi=10.1515/9781400830398.773a |isbn=978-1-4008-3039-8 |editor2-last=Barrow-Green |editor2-first=June |editor3-last=Leader |editor3-first=Imre|url-access=subscription }}</ref> who demonstrated in 1855 that matrices of this form share a property with real symmetric matrices of always having real [[Eigenvalues and eigenvectors|eigenvalues]]. Other, equivalent notations in common use are <math>A^\mathsf{H} = A^\dagger = A^\ast,</math> although in [[quantum mechanics]], <math>A^\ast</math> typically means the [[complex conjugate]] only, and not the [[conjugate transpose]].