Lyapunov matrix equation $A^HX+XA+C=O_r$ with $A$ a Jordan matrix

TL;DR

This paper provides a complete solution to Lyapunov matrix equations with Jordan matrices by reducing them to Sylvester-Lyapunov equations and solving these for Jordan blocks, advancing matrix equation theory.

Contribution

It introduces a method to convert Lyapunov equations into Sylvester-Lyapunov equations for Jordan matrices and fully solves these for Jordan blocks, enhancing understanding of matrix equations.

Findings

Complete solutions for Sylvester-Lyapunov equations with Jordan blocks.

Reduction of Lyapunov equations to Sylvester-Lyapunov form.

Explicit solutions for equations involving Jordan matrices.

Abstract

A Lyapunov matrix equation can be converted, by using the Jordan decomposition theorem for matrices, into an equivalent Lyapunov matrix equation where the matrix is a Jordan matrix. The Lyapunov matrix equation with Jordan matrix can be reduced to a system of Sylvester-Lyapunov type matrix equations. We completely solve the Sylvester-Lyapunov type matrix equations corresponding to the Jordan block matrices of the initial matrix.