How to Prove an Inequality Involving a Hermitian Negative Definite Matrix?

[marcosdnm]

Let x be in R^n and Q in Mat(R,n) where Q is hermitian and negative definite. Let (.,.) be the usual euclidian inner product.

I need to prove the following inequality:

(x,Qx) <= a(x,x)

where "a" is the maximum eigenvalue of Q.

Any idea?

 

[micromass]

Maybe try to diagonalize Q?