Hermitian matrix
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EtymologyEdit
Named after French mathematician Charles Hermite (1822–1901), who demonstrated in 1855 that such matrices always have real eigenvalues.
PronunciationEdit
NounEdit
Hermitian matrix (plural Hermitian matrixes or Hermitian matrices)
 (linear algebra) A square matrix A with complex entries that is equal to its own conjugate transpose, i.e., such that

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 equaleigenvalue 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 , then for a given state , the expectation value of the observable for that state is .
 1988, I. M. Gelfand, M. I. Graev, Geometry of homogeneous spaces, representations of groups in homogeneous spaces and related questions of integral geometry, Israel M. Gelfand, Collected Papers, Volume II, SpringerVerlag, page 366,
 There are three types of such spaces: the space of positive definite (or negative definite) Hermitian matrices, the space of nondefinite Hermitian matrices, and finally the space of degenerate Hermitian matrices p, satisfying the condition p ≥ 0 (or p ≤ 0).
 1997, A. W. Joshi, Elements of Group Theory for Physicists, New Age International, 4th Edition, page 129,
 For this we note that if H is a hermitian matrix, exp(iH) is a unitary matrix. The converse is also true, i.e., if U is any unitary matrix, then it can be expressed in the form

 U = exp(iH), (4.94)

 where H is a hermitian matrix. Now any linear combination of hermitian matrices with real coefficients is again a hermitian matrix.
 For this we note that if H is a hermitian matrix, exp(iH) is a unitary matrix. The converse is also true, i.e., if U is any unitary matrix, then it can be expressed in the form
 1998, Eugenio Iannone, Francesco Matera, Antonio Mecozzi, Marina Settembre, Nonlinear Optical Communication Networks, page 442,
 Exploiting the properties of hermitian matrixes [2], it is possible to obtain an analytical expression for the characteristic function of a hermitian quadratic form of gaussian variables, which is useful in the evaluation of transmission system performance.

SynonymsEdit
 (hermitian matrix): selfadjoint matrix
HyponymsEdit
TranslationsEdit
square matrix equal to its own conjugate transpose

