The remarks of this note are concerned with a result on transformations stated below as Theorem A, and are two-fold in nature: firstly, there are comments on the relation of this theorem to results of Perron [3] and Diliberto [l; 2], in the hope of correcting a misunderstanding that has arisen in this regard; secondly, there are remarks stressing two general properties of admissible transformat...

In this paper we derive canonical forms under structure preserving equivalence transformations for matrices and matrix pencils that have a multiple structure, which is either an H-selfadjoint or H-skew-adjoint structure, where the matrix H is a complex nonsingular Hermitian or skew-Hermitian matrix. Matrices and pencils of such multiple structures arise for example in quantum chemistry in Hartr...

The main result of this paper is the decidability of the membership problem for 2 × 2 nonsingular integer matrices. Namely, we will construct the first algorithm that for any nonsingular 2 × 2 integer matrices M1, . . . ,Mn and M decides whether M belongs to the semigroup generated by {M1, . . . ,Mn}. Our algorithm relies on a translation of numerical problems on matrices into combinatorial pro...

This note considers the problem of stability robustness with respect to a class of nonlinear time-varying perturbations which are bounded in a component-wise rather than aggregated manner. A family of robustness bounds is parameterized in terms of a nonsingular symmetric matrix. It is shown that the problem of computing the largest robustness bound over the set of nonsingular symmetric matrices...

In this paper we deal with orthogonal matrix polynomials. First of all, we establish some basic notations and results we need later. A matrix polynomial P is a matrix whose entries are polynomials, or, equivalently, a combination P(t) = A 0 +A 1 t+ +A n t n , where A 0 ; ; A n are numerical matrices (the size of all the matrices which appear in this paper is N N). A positive deenite matrix of m...

Heinig and Tewodros [18] give a set of components whose existence provides a necessary and sufficient condition for a mosaic Hankel matrix to be nonsingular. When this is the case they also give a formula for the inverse in terms of these components. By converting these components into a matrix polynomial form we show that the invertibility conditions can be described in terms of matrix rationa...

A symmetric scaling of a square matrix A #0 is a matrix of the form XAX-' where X is a nonnegative, nonsingular, diagonal matrix having the same dimension of A. An asymmetric scaling of a rectangular matrix B # 0 is a matrix of the form XBY-' where X and Y"are nonnegative, nonsingular, diagonal matrices having appropriate dimensions. We consider two objectives in selecting a symmetric scaling o...

We study the Hermitian positive definite solutions of the nonlinear matrix equation X A∗X−2A I, where A is an n × n nonsingular matrix. Some necessary and sufficient conditions for the existence of a Hermitian positive definite solution of this equation are given. However, based on the necessary and sufficient conditions, some properties and the equivalent equations of X A∗X−2A I are presented ...

In this paper we consider the parallel generalized SAOR iterative method based on the generalized AOR iterative method presented by James for solving large nonsingular system. We obtain some convergence theorems for the case when coefficient matrix is a block diagonally dominant matrix or a generalized block diagonal dominant matrix. A numerical example is given to illustrate to our results. Cr...

By a generalized inverse of a given matrix, we mean a matrix that exists for a larger class of matrices than the nonsingular matrices, that has some of the properties of the usual inverse, and that agrees with inverse when given matrix happens to be nonsingular. In theory, there are many different generalized inverses that exist. We shall consider the Moore Penrose, weighted Moore-Penrose, Draz...