Quasi-Convexity

Question

Can I get the conclusion that the function of matrix $P$ and $Q$

\begin{equation} \mathrm{tr}\left( PQ\right) \end{equation} is a quasi-concave function for $P>0$, and $Q>0$?

It is true for the scalar case $xy$ if $x>0$ and $y>0.$

Answer 1

Edit: Modified after the question was changed.

For dimensions $n\geq 2$ the statement is false:

take $P_1=Q_1=\left( \begin{array}{cc} 1 & -\frac{1}{2} \\ -\frac{1}{2} & 1 \end{array}\right)$ and $P_2=Q_2=\left( \begin{array}{cc} 1 & \frac{1}{2} \\ \frac{1}{2} & 1 \end{array}\right)$. We have

$trace(P_1Q_1)=trace(P_2Q_2)=2+\frac{1}{2} > 2 = trace\left(\frac{P_1+P_2}{2}\frac{Q_1+Q_2}{2} \right)$.

Therefore the function $f(P,Q)=trace(PQ)$ cannot be quasi-concave.