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

Abstract

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