AN ADDITION TO THE METHODS OF SOLUTION OF ONE-PARAMETRIC MATRIX PALINDROMIC PROBLEMS

Abstract

One-parameter matrix palindromic problems, as well as their transposed and conjugate analogues are considered. It is noted that the latter, through some reductions, reduce to problems of the first type, in view of which, below we consider the solution of one model example related to the problems of the first type, as the most general case. In this case, both analytical and numerical-analytical methods developed in previous articles are used.

The model example has second-order matrices, which is why the Kronecker hypermatrix generated by these matrices is a fourth-order functional matrix with a fourthdegree determinant in parameter and, therefore, four roots (singular points), two of which are zero, which use the Maclaurin approximation center for solving the problem, and the remaining two - nonzero ones that determine the use of Taylor centers for approximating the solutions to the problem.

Using the analytical method, two variants of the Kronecker hypermatrix with ranks 2 and 3 were considered at the Maclaurin and Taylor centers for approximating solutions, respectively, and a matrix approximation synthesis of one solution to the problem was carried out. Similar options were considered using the numerical-analytical method with rank 2 of the Kronecker hypermatrix at the Maclaurin center for approximation of solutions and with rank 3 of the Kronecker hypermatrix at the Taylor center of approximation of solutions. In all cases, numerous completely different solutions of the one-parameter matrix model example were obtained.

A generalization of the results shows that the considered one-parameter matrix palindromic problems in the general case have innumerable solutions, due to both the number of singular points of the corresponding Kronecker hypermatrices (solution approximation centers) and the arbitrary choice of some components of the solution hypervectors depending on the ranks of the Kronecker hypermatrices , as well as the necessary accuracy of solutions, depending on the number of matrix terms in the corresponding series of Maclaurin, Taylor, etc.