Linear matrix equations with parameters forming a commuting set of diagonalizable matrices

TL;DR

This paper establishes equivalences for the consistency of linear matrix equations with commuting diagonalizable parameters, providing solutions via the Drazin inverse and applying results to classical matrix equations.

Contribution

It introduces a theoretical framework linking matrix equation consistency, the Drazin inverse, and solution spaces for commuting diagonalizable matrices, with applications to standard equations.

Findings

Equivalence of matrix equation consistency conditions

Solution characterization using the Drazin inverse

Description of diagonalizing matrices for commuting sequences

Abstract

We prove that the following statements are equivalent: a linear matrix equation with parameters forming a commuting set of diagonalizable matrices is consistent, a certain matrix constructed with the Drazin inverse is a solution of this matrix equation, the attached standard linear matrix equation is consistent. The number of zero components of a given matrix (the relevant matrix) gives us the dimension of the affine space of solutions. We apply this theoretical results to the Sylvester equation, the Stein equation, and the Lyapunov equation. We present a description of the set of the diagonalizing matrices for a commuting sequence of diagonalizable matrices.