Eigenvalues of a matrix involved in a linear matrix equality

**Riesz** · November 27th 2010, 02:36 AM

We know that:

with

,

e

: matrixes $\in\Re^{nxn}$ ;

: generic eigenvalue of A;

To be proved:

with

: positive scalar;

Writing

with

,

we can prove

, because

.

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

**Opalg** · November 27th 2010, 11:05 AM

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

**Riesz** · November 27th 2010, 03:07 PM

You are right, good suggestion!

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