Hermitian matrix

EnglishEdit

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EtymologyEdit

Named after Charles Hermite (1822–1901), French mathematician.

PronunciationEdit

• "her mission matrix"
• (US) IPA(key): /hɝ.ˈmɪ.ʃən ˈmeɪ.tɹɪks/

NounEdit

Hermitian matrix ‎(plural Hermitian matrices)

1. (linear algebra) a square matrix with complex entries that is equal to its own conjugate transpose, i.e., a matrix such that ${\displaystyle A=A^{\dagger }\,,}$ where ${\displaystyle A^{\dagger }}$ denotes the conjugate transpose of a matrix A
Hermitian matrices have real diagonal elements as well as real eigenvalues.[1]
If a Hermitian matrix has a simple spectrum (of eigenvalues) then its eigenvectors are orthogonal.[2] On the other hand, a set of two or more eigenvectors with the same eigenvalue can be orthogonalized (e.g., through the Gram–Schmidt process, since any linear combination of equal-eigenvalue eigenvectors will also be an eigenvector) and will already be orthogonal to other eigenvectors which have different eigenvalues.
If an observable can be described by a Hermitian matrix ${\displaystyle H}$, then for a given state ${\displaystyle |A\rangle }$, the expectation value of the observable for that state is ${\displaystyle \langle A|H|A\rangle }$.