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

2024/05/06 by Dan Comănescu, Comănescu, Dan
Computer Science · #15A06 #15A09 #15A10 #15A20 #15A24 #15A27 #FOS: Mathematics #Matrix Theory and Algorithms #Rings and Algebras (math.RA)

paper · pdf · doi:10.48550/arxiv.2406.09429

openalex publication_date 2024/05/06 · openalex created_date 2024/06/18 · openalex updated_date 2026/07/28

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.

Related

• The matrix equation XA-AX=f(X) when A is diagonalizable
• Explicit formulas for determinantal representations of the Drazin\n inverse solutions of some matrix and differential matrix equations
• Anti-commuting Solutions of the Yang-Baxter-like Matrix Equation
• A note on solutions of linear systems
• On the solution of the linear matrix equation X=Af(X)B+C