Eigenvalues of a matrix involved in a linear matrix equality

We know that:

http://www.thescienceforum.com/latex...9933e69196.png http://www.thescienceforum.com/latex...2a6f5eba80.png http://www.thescienceforum.com/latex...55605557ff.png

with http://www.thescienceforum.com/latex...b72eacbe29.png, http://www.thescienceforum.com/latex...319e541aee.png e http://www.thescienceforum.com/latex...0c9a0fdaaa.png: matrixes $\in\Re^{nxn}$ ; http://www.thescienceforum.com/latex...391549a107.png: generic eigenvalue of A;

To be proved:

http://www.thescienceforum.com/latex...9933e69196.png http://www.thescienceforum.com/latex...c7a8d83d9e.png http://www.thescienceforum.com/latex...55605557ff.png

with http://www.thescienceforum.com/latex...38408b5f33.png: positive scalar;

Writing http://www.thescienceforum.com/latex...b919a71b61.png with http://www.thescienceforum.com/latex...ac1b972c23.png,

we can prove http://www.thescienceforum.com/latex...544b079836.png http://www.thescienceforum.com/latex...55605557ff.png, because http://www.thescienceforum.com/latex...08ebe2cbe8.png.

How to link eigenvalues of A to $\overline{Q}$ ?

How about taking the "known" result (for matrices satisfying $A^{\textsc t}P + PA = -Q$ ), and applying it with $A+cI$ in place of $A$ ?

You are right, good suggestion!