Linear Algebra and Its Applications · Problem 6.2.18

Positive Definite Matrices — Gilbert Strang

Mathematics · tests-for-positive-definiteness

Problem

(Lyapunov test for stability of

M

) Suppose

AM + M^H A = -I

with positive definite

A

. If

Mx = \lambda x

show that

\mathrm{Re}\,\lambda < 0

. (Hint: Multiply the first equation by

x^H

and

x