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
2026-04-12 22:54:06.1776034446
properties of largest eignvalue of product of two matrices
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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 $x' B x \le \lambda_\max x' A x$.