hermitian matrices equations theorem

Question

Let $A,B$ be square matrices. We define $A_h=0.5(A+A^*)$, $\;B_h=0.5(B+B^*)$ their hermitian part.

We have to prove that $A_h=B_h => x^*Ax=x^*Bx$

it seems trivial but it isn't..

Is the other direction also correct?

Answer 1

If $A_h - B_h = 0$, $A - B$ is skew-Hermitian. Then $x^*(A-B) x = x^* (A-B)^* x = -x^* (A-B) x$ so $x^* (A-B) x = 0$.

Answer 2

thank you so much for your quick answer.

however, this equation does not make any sence to me since $A-B$ is skew hermitian $x^∗ (A−B)x = x^∗ (A−B)^* x$

can you explain that? is it possible that the proof is not correct on this side => and only on the other one <= ?