equivalence of quadratic forms

Question

Given two Hermitian positive semidefinite matrices $A$ and $B$, under what conditions on these matrices will $x^H A x = x^H B x$ for all vectors $x$?

Clearly, we have equivalence when $A=B$, but I feel like this is too stringent a condition, because I do not require $y^H Ax = y^H Bx$ for all $x$ and $y$.

Answer 1

If $C$ is a Hermitian matrix then $C$ is diagonalizable. If $c$ is an eigenvalue of $C$ associated to an normalized eigenvector $x$ then $x^HCx=c$. So if $x^HCx=0$ for every $x$ then every eigenvalue of $C$ is zero. Since $C$ is diagonalizable then $C=0$.

Now put $C=A-B$. Therefore $A=B$.