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I have seen somewhere the following results related to Lyapunov equation:

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.

(1) A necessary and sufficient condition for $\Re\lambda(A) > \lambda_1$ is that there exists a symmetric and positive definite matrix P soluting $PA+A^TP > 2\lambda_1 P$;

(2) A necessary and sufficient condition for $\Re\lambda(A) < \lambda_2$ is that there exists a symmetric and positive definite matrix P soluting $PA+A^TP < 2 \lambda_2 P$.

Can someone give me some clues or references to the proofs? Thanks

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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.

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