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 ?
Because Hermitian matrices have only real eigenvalues and since (also by using techniques from [7]) for a given such matrix A and a given real [lambda], we succeed in evaluating all the polynomials in the sequence (1.
Here, we present an explicit construction of hives from Hermitian matrices whose proofs require little more than linear algebra.
1] be the Jordan algebra of r x r Hermitian matrices over R with composition A [omicron] B = 1/2(AB + BA) and let [SIGMA] [epsilon] [H.
It follows from Weyl's inequality that if A and B are Hermitian matrices, then
A relevant aspect of [2, 3] is the surrounding of the desired point by four points on the FOV boundary determined by considering the largest and the smallest eigenvalues of the Hermitian matrices H ([e.
Hence, the law of inertia yields a tool to locate eigenvalues of Hermitian matrices or definite matrix pencils.
Psarrakos [42] used this characterization to propose an algorithm to determine if a triple of Hermitian matrices is definite.
Other methods with the same mathematical properties are the Generalized Minimum Residual (GCR) method [17], the MinRes algorithm of Paige and Saunders [49] for Hermitian matrices, and, the Conjugate Residual (CR) method of Stiefel [59] for Hpd matrices.
In the following, the notation A [less than or equal to] B between Hermitian matrices A and B is meant to be with respect to the cone of positive-semidefinite matrices, i.
4) shows that the Hermitian matrices B and D are congruent, and therefore they must have the same inertia.
GOLUB, On the spectral decomposition of Hermitian matrices modified by low rank perturbations with applications, SIAM J.
We have given a convergence analysis of the restarted Arnoldi approximation for functions of Hermitian matrices in the case when the restart length is one.