# Hermitian matrix

## EnglishEdit

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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

• (US) IPA(key): /hɝ.ˈmɪ.ʃən ˈmeɪ.tɹɪks/

### NounEdit

1. (linear algebra) A square matrix A with complex entries that is equal to its own conjugate transpose, i.e., such that ${\displaystyle A=A^{\dagger }.}$
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]
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 ${\displaystyle \langle A|H|A\rangle }$ .
• 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, Springer-Verlag, 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.
• 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.