Is it true that when $X$ is a positive matrix and $Y$ is a self adjoint matrix then $XY$ is positive??

Question

Let $X_{n \times n}$ be a positive matrix i.e $<Xy,y> \ge 0$ for all $y \in \mathbb{C^n}$ and $Y_{n \times n}$ be a self adjoint matrix. Show that $XY$ is positive or find a counterexample.

So I look at $<XYz,z>=<\sqrt{X}Yz,\sqrt{X}z>$

Had $YX=XY$,then $\sqrt{Y}X=X\sqrt{Y}$ which would give $<XYz,z>=<\sqrt{X}Yz,\sqrt{X}z>=<Y\sqrt{X}z,\sqrt{X}z> \in \mathbb{R}$ as $Y$ is a selfadjoint matrix. But then this is no conclusion even when $X$ and $Y$ commute.

Is it even true?? Any hints??

Thanks for the help!!

Answer 1

Hint: Consider the case $n = 1$.