The Matrix of Linear Mappings

TL;DR

This paper explores the algebraic structure of matrices composed of linear mappings within an associative algebra, defining matrix operations, inverse conditions, and methods for solving linear systems in this context.

Contribution

It introduces a framework for matrices of linear mappings over associative algebras, including defining multiplication, inverse conditions, and solution methods for linear systems.

Findings

Defined matrix product of linear mappings

Established conditions for invertibility of such matrices

Presented methods for solving linear equations in associative algebras

Abstract

On the set of mappings of the given set, we define the product of mappings. If A is associative algebra, then we consider the set of matrices, whose elements are linear mappings of algebra A. In algebra of matrices of linear mappings we define the operation of product. The operation is based on the product of mappings. If the matrix a of linear mappings has an inverse matrix, then the quasideterminant of the matrix a and the inverse matrix are matrices of linear mappings. In the paper, I consider conditions when a matrix of linear mappings has inverse matrix, as well methods of solving a system of linear equations in an associative algebra.