An eigen decomposition based closed-form solution for the Discrete Lyapunov and Stein Equations

TL;DR

This paper introduces an eigen decomposition-based closed-form solution for the discrete Lyapunov and Stein equations, offering an efficient alternative to iterative methods for certain matrix families.

Contribution

It provides a new explicit solution for the DLE and Stein equations using eigen decomposition, applicable to specific matrix families and general matrices when ED is computable.

Findings

Closed-form solution for DLE using eigen decomposition.

Extension of solution to Stein equation.

Comparable complexity to iterative methods, simpler than existing closed-form solutions.

Abstract

A simple closed-form solution to the discrete Lyapunov equation (DLE) is established for certain families of matrices. This solution is expressed in terms of the eigen decomposition (ED) for which closed-form solutions are known for all 2x2; 3x3 and certain families of matrices. For general matrices, the proposed ED based closed-form solution can be used as an efficient numerical solution when the ED can be computed. The result is then extended to give closed-form solutions for a generalization of the DLE, called the Stein equation. The proposed explicit solution's complexity is of the same order as iterative solutions and significantly smaller than known closed-form solutions. These solutions may prove convenient for analysis and synthesis problems related to these equations due to their compact form.