Justification of conditions for unique solvability of matrix equations with two triangular unknowns and mutually inverse coefficients

Abstract

In the article the object of research are the matrix equations. The role of the matrix and the matrix equations in the theoretical and practical issues is well known. In its simplest form it arises in different theoretical and applied problems related to the solution of systems of linear algebraic equations. For example, in mechanics, physics, electrical engineering, hydraulics, economy.

A unique solvability of the two abstract matrix equations is investigated for the next form:

                                                 AX⁺+Y_–=B,                                             (1)

                                               A^-1X⁺₁+Y_1–=B,                                          (2)

with unknown lower X⁺, X⁺₁ and the upper Y_–, Y_1–, triangular matrices and mutually inverse matrices – the coefficients A, A^-1. The approach is based on the interpretation of equations (1), (2) as the implementations in the ring of matrices of corresponding equations in the abstract ring with a pair of factorization, based on the basic provisions of the theory of rings and operators. In particular, special developed projections are used. It is characterized by significantly less than the maximum order of determinants of matrices, which have to operate using the proposed approach and its results. It is substantially less than the orders of the determinants that arise in the transition from (1), (2) to systems of linear algebraic equations by equating the corresponding matrix elements in their left and right sides.

The theorem on the unique solvability of these equations with matrix representations of the solutions is formulated and proved, which gives an accurate method for solving specific equations (1), (2) and their corresponding tasks.

An illustrative example is given.