I encountered the following question from Problems and Solutions in Introductory and Advanced Matrix Calculus by Steeb and Hardy (P.39 Problem 3).
Question. Let $A\in M_n(\mathbb{C})$ be a positive semi-definite matrix, and $u,v\in\mathbb{C}^n$ two unit vectors. Prove that $$(u^*v)(u^*Av)\ge 0.$$
However, I do not think this is true. For $n=1$, we have $A=[a]$ for some non-negative real number $a$. Write $u=[e^{i\theta}]$ and $v=[e^{i\phi}]$ for some $\theta,\phi\in(-\pi,\pi]$. Then $$(u^*v)(u^*Av)=ae^{i\cdot 2(\phi-\theta)}, $$ which needs not be a real number.
I am wondering if this inequality is true if the conditions are slightly modified. Any suggestions or comments are highly welcomed.
This exercise does not look correct. In fact, even if you pick $A=\alpha I$, $\alpha>0$, then you would get $\alpha(u*v)^2$ which is not necessarily real.
Considering instead $(v^*u)(u^*Av)$ fixes the problem in that case but does not generalize to other positive definite matrices.
I have been trying to fix the statement of the problem, to no avail. I will update my answer if I manage to come up with a slight modification that makes the problem meaningful.