Using the notions of the numerical range, Schur complement and unitary equivalence, an eigenvalue inequality is obtained for a general complex matrix, giving rise to a region in the complex plane that contains its spectrum. This region is determined by a curve, generalizing and improving classical eigenvalue bounds obtained by the Hermitian and skew-Hermitian parts, as well as the numerical ran...

In this paper, we present new results relating the numerical range of a matrix A with generalized Levinger transformation L(A, α, β) = αH A + βS A , where H A and S A , are, respectively the Hermitian and skew-hermitian parts of A. Using these results, we, then derive expressions for eigenvalues and eigenvectors of the perturbed matrix A + L(E, α, β), for a fixed matrix E and α, β are real para...

Using the notions of the numerical range, Schur complement and unitary equivalence, an eigenvalue inequality is obtained for a general complex matrix, giving rise to a region in the complex plane that contains its spectrum. This region is determined by a curve, generalizing and improving classical eigenvalue bounds obtained by the Hermitian and skew-Hermitian parts, as well as the numerical ran...

The optimal parameter of the Hermitian/skew-Hermitian splitting (HSS) iteration method for a real 2-by-2 linear system is obtained. The result is used to determine the optimal parameters for linear systems associated with certain 2-by-2 block matrices, and to estimate the optimal parameters of the HSS iteration method for linear systems with n-by-n real coefficient matrices. Numerical examples ...

We consider the iterative solution of weighted Toeplitz least squares problems. Our approach is based on an augmented system formulation. We focus our attention on two types of preconditioners: a variant of constraint preconditioning, and the Hermitian/skew-Hermitian splitting (HSS) preconditioner. Bounds on the eigenvalues of the preconditioned matrices are given in terms of problem and algori...

A generalization of the cyclic Jacobi algorithm is proposed that works in an arbitrary compact Lie algebra. This allows, in particular, a unified treatment of Jacobi algorithms on different classes of matrices, such as, e.g., skew-symmetric or skew-Hermitian Hamiltonian matrices. Wildberger has established global, linear convergence of the algorithm for the classical Jacobi method on compact Li...

We derive a formula for the backward error of a complex number λ when considered as an approximate eigenvalue of a Hermitian matrix pencil or polynomial with respect to Hermitian perturbations. The same are also obtained for approximate eigenvalues of matrix pencils and polynomials with related structures like skew-Hermitian, ∗-even and ∗-odd. Numerical experiments suggest that in many cases th...

In this paper, we apply the generalized Hermitian and skew-Hermitian splitting (GHSS) iterative method to the problem of image restoration. We use a new splitting of the Hermitian part of the coefficient matrix of the problem. Moreover, we introduce a restricted version of the GHSS (RGHSS) iterative method together with its convergence properties. The optimal parameter, which minimizes the spec...

In this paper, a modified Hermitian and skew-Hermitian splitting (MHSS) iteration method for solving the complex linear matrix equation AXB = C has been presented. As the theoretical analysis shows, the MHSS iteration method will converge under certain conditions. Each iteration in this method requires the solution of four linear matrix equations with real symmetric positive definite coefficien...

In this paper, we study the structure of cyclic, quasi-cyclic, constacyclic codes and their skew codes over the finite ring R = Z3+vZ3+v 2 Z3, v 3 = v. The Gray images of cyclic, quasi-cyclic, skew cyclic, skew quasicyclic and skew constacyclic codes over R are obtained. A necessary and sufficient condition for cyclic (negacyclic) codes over R that contains its dual has been given. The paramete...