I have been working through Roger A. Horn and Charles R. Johnson's Matrix Analysis book and have come stuck on one direction of an if and only if proof. It goes as follows;
Suppose $A, B \in \mathbb{C}^{n\times n}$ are Hermitian. Show that,
$x^*Ax = x^*Bx \implies A = B$
My thought process is as follows,
$x^*Ax = x^*Bx$
$\implies x^*(A-B)x = 0$
as this must hold for all $x$ then
$A=B$
Any help would be appreciated!
You may make use of the polarization identities
\begin{aligned} \Re(y^\ast Mx)&=\frac14\left[(x+y)^\ast M(x+y) - (x-y)^\ast M(x-y)\right],\\ \Im(y^\ast Mx)&=\frac14\left[(x-iy)^\ast M(x-iy) - (x+iy)^\ast M(x+iy)\right]. \end{aligned}