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



Eigenvalue estimation by Lyapunov's method
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. 


