Eigenvalue estimation by Lyapunov's method - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T14:39:56Z http://mathoverflow.net/feeds/question/96427 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/96427/eigenvalue-estimation-by-lyapunovs-method Eigenvalue estimation by Lyapunov's method W. Nyway 2012-05-09T10:39:58Z 2012-05-09T14:20:59Z <p>I have seen somewhere the following results related to Lyapunov equation:</p> <p>Let $A\in \mathbb{R}^n$ be a stable matrix in the sense of having negative real part eigenvalues. Let $\Re\lambda()$ denote the real part of a eigenvalu of a matrix.</p> <p>(1) A necessary and sufficient condition for $\Re\lambda(A) > \lambda_1$ is that there exists a symmetric and positive definite matrix P soluting $PA+A^TP > 2\lambda_1 P$;</p> <p>(2) A necessary and sufficient condition for $\Re\lambda(A) &lt; \lambda_2$ is that there exists a symmetric and positive definite matrix P soluting $PA+A^TP &lt; 2 \lambda_2 P$.</p> <p>Can someone give me some clues or references to the proofs? Thanks</p> http://mathoverflow.net/questions/96427/eigenvalue-estimation-by-lyapunovs-method/96454#96454 Answer by Michael Renardy for Eigenvalue estimation by Lyapunov's method Michael Renardy 2012-05-09T14:20:59Z 2012-05-09T14:20:59Z <p>Hale, Ordinary Differential Equations, Lemma 1.5 in Chapter X. This refers to the 1980 edition. It is the chapter on Liapunov's direct method. </p>