Let $X$ and $Y$ be positive semi-definite self-adjoint complex matrices of same finite order. The, is it true that $|X-Y|\leq X+Y$ where for any matrix $A$, $|A|$ is defined to be $|A|:=(A^\*A)^{\frac{1}{2}}$ ?


PS. I think the answer is **No**. But I could not find any counterexample!



A Matrix Inequality for positive definite matrices
The answer is **No**. Here is a counter-example:
$$X=\begin{pmatrix} 9 & 3 \\ 3 & 1 \end{pmatrix},\qquad Y=\begin{pmatrix} 1 & 3 \\ 3 & 9 \end{pmatrix}.$$


