Appendix: Matrix Equations

Jan H. van Schuppen*

*Corresponding author for this work

Research output: Chapter in Book/Conference proceedings/Edited volumeChapterScientific


The Lyapunov equation and the algebraic Riccati equation are treated in depth. The Lyapunov equation arises as the equation for the asymptotic covariance matrix of the state of a stationary Gaussian system. The algebraic Riccati equation arises in the Kalman filter, in stochastic control, and in stochastic realization of a Gaussian system. Results for both equations are provided on: the existence of a solution, uniqueness with respect to conditions, a description of the set of all solutions if applicable, particular properties of solutions, and on the computation of solutions.

Original languageEnglish
Title of host publicationControl and System Theory of Discrete-Time Stochastic Systems
Number of pages43
ISBN (Electronic)978-3-030-66952-2
Publication statusPublished - 2021

Publication series

NameCommunications and Control Engineering
ISSN (Print)0178-5354
ISSN (Electronic)2197-7119


  • Algebraic Riccati equation
  • Lyapunov equation


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