properties of largest eignvalue of product of two matrices

Question

I'm searching for the proof of this lemma it's about largest eignvalue of product of two matrices. one of them is positive definete and the other one is symmetric. B is symmetric matrix, A is Positive definite. Then : x'Bx =< landamax(inv(A)B).x'Ax

Answer 1

If $B$ is symmetric and $A$ positive definite, let $C = A^{-1/2} B A^{-1/2}$. Then $C$ has the same eigenvalues as $A^{-1} B$. Moreover, for any $x$, if $y = A^{1/2} x$ we have $x' A x = y' y$ and $x' B x = y' C y$. If $\lambda_\max$ is the largest eigenvalue of $C$, then $y' C y \le \lambda_\max \; y' y$. Put it all together and you have $x' B x \le \lambda_\max x' A x$.