Hermitian matrix

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English[edit]

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Etymology[edit]

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

Pronunciation[edit]

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

Noun[edit]

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 A = A^\dagger\,, where 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 H, then for a given state |A\rangle, the expectation value of the observable for that state is \langle A|H|A\rangle.

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References[edit]

  1. ^ Proof Wiki — Hermitian Operators have Real Eigenvalues
  2. ^ Proof Wiki — Hermitian Operators have Orthogonal Eigenvectors