Hermitian matrix

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Hermitian matrix

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• hermitian matrix

Named after French mathematician Charles Hermite (1822–1901), who demonstrated in 1855 that such matrices always have real eigenvalues.

• (US) IPA^(key): /hɝˈmɪ.ʃən ˈmeɪ.tɹɪks/

Hermitian matrix (plural Hermitian matrixes or Hermitian matrices)

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 ${\displaystyle \langle A\rangle }$, 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.

• normal matrix

• Pauli matrix
• Gramian matrix
• self-adjoint matrix
• symmetric matrix, real matrix