I want to know the correction of the following inequality:
If the matrix $A$ is (real, symmetric) negative definite, and a given real vector $u \neq 0$, is there a real number $c_1$ such that
$u^T Au \leq c_1u^Tu < 0$.
I think yes, $-c_1$ can be the smallest eigenvalue of the matrix $-A$. However, can we give a specified bound for the $c_1$? or are there some references for this topic?