I'm having a little difficulty with this. Given some positive operator $A$, show that it is also hermitian.

(A positive operator is defined as $\langle Ax,x\rangle\ge 0$ for all $x \in V$ where $V$ is some vector space.)

Here's what I have so far.
We can construct $A = B + iC$ where $B,C$ are hermitian operators
$B = (A + A^\*)/2$, $C = (-iA + iA^\*)/2$ where $^\*$ is the conjugate transpose.

I'm trying to show that $B$ and $C$ are diagonalizable by the same vectors, and that the eigenvalues of $C$ are $0$. I'm not sure how to do this though.