Hermitian Matrix


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Hermitian matrix

[er′mish·ən ′mā·triks]
(mathematics)
A matrix which equals its conjugate transpose matrix, that is, is self-adjoint.

Hermitian Matrix

 

(or self-adjoint matrix), a matrix coincident with its adjoint, that is, a matrix such that aik= āki, where ā is the complex conjugate of the number a. If the elements of a Hermitian matrix are real, then the matrix is symmetric. A Hermitian matrix has real eigenvalues λ1, λ2, …, λn and corresponds to a linear transformation in a complex n-dimensional space that reduces to stretchings by ǀλiǀ in n mutually perpendicular directions and reflections in the planes orthogonal to the directions for which λi < 0. A bilinear form

whose coefficients form a Hermitian matrix, is called a Hermitian form. Any matrix can be written in the form A1 + iA2, where A1 and A2 are Hermitian matrices, and in the form A ∪, where A is a Hermitian matrix and U is a unitary matrix. If A and B are Hermitian matrices, then A B is a Hermitian matrix if and only if A and B commute.

References in periodicals archive ?
According to Lemma 5 with a positive definite Hermitian matrix P, a symmetric form is presented as follows.
Furthermore, let [sigma](M) denote the spectrum of a Hermitian matrix M.
p] is a Hermitian matrix, its eigenvalues are real and corresponding eigenvectors are orthogonal with each other.
Keywords Generalized Hermitian matrix, full-rank factorization, Procrustes problem, optimal approximation.
The analysis is general for any linear operator D which could be modeled with a Hermitian matrix and is not limited to sampling problems.
On the spread of a hermitian matrix and a conjecture of thompson.
If A is an n x n Hermitian matrix, then it is well known that all the eigenvalues of A are real.
Unfortunately, the Green's function matrix has a large condition number (on the order of 10(10)), so when the Hermitian matrix is formed from the Green's function matrix (see the procedure in Mautz (3) and Hill (4)), the condition number is even higher (on the order of 10 (17)).
We take the pattern BCSPWR10 and fill-in a Hermitian matrix.
Since all column and row determinants of a Hermitian matrix over H are equal, we can define the determinant of a Hermitian matrix A [member of] M(n, H).