Two Strange Facts

Here are two strange facts about matrices, which I can prove but not in a satisfying way.

  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 pth 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?

If anyone has insight into either of these, I’d be very interested!

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