Toeplitz Matrices in the Problem of Semiscalar Equivalence of Second-Order Polynomial Matrices

B Z Shavarovskii

doi:10.1155/2017/6701078

Abstract

We consider the problem of determining whether two polynomial matrices can be transformed to one another by left multiplying with some nonsingular numerical matrix and right multiplying by some invertible polynomial matrix. Thus the equivalence relation arises. This equivalence relation is known as semiscalar equivalence. Large difficulties in this problem arise already for 2-by-2 matrices. In this paper the semiscalar equivalence of polynomial matrices of second order is investigated. In particular, necessary and sufficient conditions are found for two matrices of second order being semiscalarly equivalent. The main result is stated in terms of determinants of Toeplitz matrices.

Highlights

Let C be a field of complex numbers and C[x] the ring of polynomials in an indeterminate x over C
Toeplitz matrices plays an important role in the conditions under which two matrices of second order can be transformed to one another by semiscalar equivalent transformation
Any matrix has connection with Toeplitz matrices in the sense that every matrix can be represented as a sum of the products of Toeplitz matrices

Summary

Introduction

Let C be a field of complex numbers and C[x] the ring of polynomials in an indeterminate x over C. Has attracted much attention for many years This proved to be a bigger problem than originally anticipated. The matrices F(x), G(x) ∈ M(n, C[x]) are called semiscalarly equivalent, if the equality (1) is satisfied for some nonsingular matrix P ∈ M(n, C) and for some invertible matrix Q(x) ∈ M(n, C[x]) [1] (see [2]). Due to this fact the problem of finding the conditions under which the matrices are semiscalarly equivalent is of current interest. Toeplitz matrices plays an important role in the conditions under which two matrices of second order can be transformed to one another by semiscalar equivalent transformation. The results of this paper may be applied in the solving of the matrix equations, which are utilized in many problems of engineering

Preliminaries

The Case 1

The Case 2

Auxiliary Statements

The Case 3