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 is also not true in general in the $2\times 2$ case: consider for instance 
$$
L_1=\begin{bmatrix} 3&0\\0&0\end{bmatrix},\ \ \ \ \ L_2=\begin{bmatrix}4&2\\2&4\end{bmatrix}.
$$
Then
$$
L_2^2-L_1^2=\begin{bmatrix} 11&16\\16&20\end{bmatrix},
$$
which is not positive semidefinite. 
