Conditions of Semiscalar Equivalence of One Class 3×3 Matrices of Simple Structure

B Z Shavarovskii,Serkan Araci

doi:10.1155/2022/8395922

Abstract

This paper solves the problem of classifying matrices over a ring of polynomials up to semiscalarly equivalence transformations. For the selected set of polynomial matrices of simple structure, the so-called oriented by characteristic roots reduced matrix is constructed. The latter, in addition to the triangular form and the presence of invariant factors on the main diagonal, has some predefined properties. Invariants and conditions of semiscalar equivalence are established for such matrices.

Highlights

A similar result was obtained in [3, 4]. e specified triangular matrix is defined by the class {PF(x)Q(x)} ambiguously; it cannot be used at once for the establishment of belonging of any matrix to this class. us, there is a need to specify in class {PF(x)Q(x)} such a matrix, which is determined with a lesser degree of ambiguity and for which we can find a system of invariants
Matrix B(x) of Proposition 8 is called oriented by the characteristic roots α0, α1 ∈ M1, β0 ∈ M2 reduced matrix
In the class of semiscalarly equivalent polynomial matrices of simple structure, the so-called oriented by characteristic roots reduced matrix is established

Summary

Introduction

Let C be the field of complex numbers and C[x] the ring of polynomials in an indeterminate x over C. Erefore, there is a problem of classification of polynomial matrices up to semiscalar equivalence. E problem of classifying of polynomial matrices of the same size up to semiscalar equivalence contains the matrix pair problem, which is the problem of classifying a pair of square matrices of the same size up to similarity transformations. Us, the problem of classifying polynomial matrices up to semiscalar equivalence is wild. We investigate the problem of determining when two matrices are semiscalarly equivalent. Much useful information on the issues discussed in this article can be found in the monograph [8] This monograph investigates other types of equivalences of matrices and finite sets of matrices over different rings. Some generalizations of the concept of semiscalar equivalence in the case of matrices over quadratic rings and their application to the solution of matrix equations can be found in recent work [9]

Preliminary Information

Reduction to a Special Triangular Form

Main Results

Conclusion