Squares of positive semidefinite matrices Suppose $L_1  \succeq L_2$, where $L_1,L_2$ are positive semidefinite matrices (actually combinatorial Laplacians). Is the following inequality true, and if no, under which conditions?
$$L_1^2  \succeq L_2^2$$

It's not always true. Counterexample:
$$
\begin{align*}
&L_1=\pmatrix{1&-1&0\\ -1&2&-1\\ 0&-1&1},
\ L_2=\pmatrix{1&-1&0\\ -1&1&0\\ 0&0&0},\\
&L_1^2-L_2^2=\pmatrix{0&-1&1\\ -1&4&-3\\ 1&-3&2}.
\end{align*}
$$
I'm not sure if there is any good (non-restrictive) sufficient condition for the inequality to hold.
