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 ?
Furthermore, let [sigma](M) denote the spectrum of a Hermitian matrix M.
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.
The GUE random matrix ensemble is a probability measure on the set of k x k Hermitian matrix with density proportional to exp(-Trace([X.
The Hermitian matrix polynomial Q([lambda]) is definite if and only if any two (and hence all) ofthe followingproperties hold:
Therefore, an alternative approach is to show that every Hermitian matrix contained in an interval matrix Y [contains or equal to] [?
Then the Hermitian matrix [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.
are the entries of the Jacobi matrix of the measure v associated with the Hermitian matrix [OMEGA].
n]) is the maximal eigenvalue of the Hermitian matrix Re [A.
max](x) denotes the maximum eigenvalue of a Hermitian matrix.
linear systems, two-stage methods, block methods, multisplitting methods, Hermitian matrix, positive definite matrix, preconditioners, parallel algorithms, monotonicity, distributed memory.