Q:

Eigenvalue estimation by Lyapunov's method

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

A:

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. 

