Khatri and Mitra [3] gave necessary and sufficient conditions for the existence of the common Hermitian solution to (2) and presented an explicit expression for the general Hermitian solution to (2) by generalized inverses. Using the singular value decomposition (SVD), Yuan [4] investigated the general symmetric solution of (2) over the real number field R. By the SVD, Dai and Lancaster [5] con...

We present eigenvalue bounds for perturbations of Hermitian matrices, and express the change in eigenvalues in terms of a projection of the perturbation onto a particular eigenspace, rather than in terms of the full perturbation. The perturbations we consider are Hermitian of rank one, and Hermitian or non-Hermitian with norm smaller than the spectral gap of a specific eigenvalue. Applications ...

By using the complex representation of quaternion matrices, the Moore–Penrose generalized inverse and the Kronecker product of matrices, the expressions of the least squares η-Hermitian solution with the least norm and the expressions of the least squares η-anti-Hermitian solution with the least norm are derived for the matrix equation AXB+CXD = E over quaternions.

Lavrentiev regularization is a popular approach to the solution of linear discrete illposed problems with a Hermitian positive semidefinite matrix. This paper describes Lavrentiev-type regularization methods that can be applied to the solution of linear discrete ill-posed problems with a general Hermitian matrix. Fractional Lavrentiev-type methods as well as modifications suggested by the solut...

Let R be an n by n nontrivial real symmetric involution matrix, that is, R R−1 R / In. An n × n complex matrix A is termed R-conjugate if A RAR, where A denotes the conjugate of A. We give necessary and sufficient conditions for the existence of the Hermitian R-conjugate solution to the system of complex matrix equations AX C andXB D and present an expression of the Hermitian R-conjugate soluti...

We prove a convergence result for an iterative method, proposed recently by B. Meini, for finding the maximal Hermitian positive definite solution of the matrix equationX+A∗X−1A = Q, where Q is Hermitian positive definite.

This paper proposes neural network for computing the eigenvectors of Hermitian matrices. For the eigenvalues of Hermitian matrices, we establish an explicit representation for the solution of the neural network based on Hermitian matrices and analyze its convergence property. We also consider to compute the eigenvectors of skew-symmetric matrices and skew-Hermitian matrices, corresponding to th...

C06FKF Circular convolution or correlation of two real vectors, extra workspace for greater speed C06FPF Multiple one-dimensional real discrete Fourier transforms C06FQF Multiple one-dimensional Hermitian discrete Fourier transforms C06FRF Multiple one-dimensional complex discrete Fourier transforms C06FUF Two-dimensional complex discrete Fourier transform C06FXF Three-dimensional complex discr...

Toeplitz matrices have found important applications in bioinformatics and computational biology [5, 6, 11, 12]. In this paper we concern the spectral properties of hermitian Toeplitz matrices. Based on the fact that every centrohermitian matrix can be reduced to a real matrix by a simple similarity transformation, we first consider the eigenstructure of hermitian Toeplitz matrices and then disc...

Left and right inverse eigenpairs problem for κ-hermitian matrices and its optimal approximate problem are considered. Based on the special properties of κ-hermitian matrices, the equivalent problem is obtained. Combining a new inner product of matrices, the necessary and sufficient conditions for the solvability of the problem and its general solutions are derived. Furthermore, the optimal app...