# Two Strange Facts

1. If $A$ and $B$ are symmetric matrices satisfying $0 \preceq A \preceq B$, then $A^{1/2} \preceq B^{1/2}$, and $B^{-1} \preceq A^{-1}$, but it is NOT necessarily the case that $A^2 \preceq B^2$. Is there a nice way to see why the first two properties should hold but not necessarily the third? In general, do we have $A^p \preceq B^p$ if $p \in [0,1]$?
2. Given a rectangular matrix $W \in \mathbb{R}^{n \times d}$, and a set $S \subseteq [n]$, let $W_S$ be the submatrix of $W$ with rows in $S$, and let $\|W_S\|_*$ denote the nuclear norm (sum of singular values) of $W_S$. Then the function $f(S) = \|W_S\|_*$ is submodular, meaning that $f(S \cup T) + f(S \cap T) \leq f(S) + f(T)$ for all sets $S, T$. In fact, this is true if we take $f_p(S)$, defined as the sum of the $p$th powers of the singular values of $W_S$, for any $p \in [0,2]$. The only proof I know involves trigonometric integrals and seems completely unmotivated to me. Is there any clean way of seeing why this should be true?