Two Strange Facts

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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 $p$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?

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

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Shen

Hi Jacob,

I was just reading this post by Terrance Tao https://terrytao.wordpress.com/2017/05/22/quantitative-continuity-estimates/ and the last few paragraphs reminded me of the first fact stated here. My gut feeling is the Sylvester equation based approach might be helpful in establishing true/false conclusion of the ‘In general…’ question you asked in the end.

jsteinhardt

Thanks a lot! I’ll take a look.

Best, Jacob

Mark Sellke

Functions with the “operator monotone property” in 1. are worked out in some detail in this survey paper: https://arxiv.org/pdf/1305.2471.pdf. Key theorems are 2.3, which shows that it does hold for all p in [0,1], and 4.1, which classifies the full set of (positive, though this doesn’t really matter) functions that work as weighted averages of the functions f_t(x)=x/(x+t) by a finite measure (so these x/(x+t) functions are the extreme points). Here f_0(x)=1 and f_{\infty}(x)=x.

A corollary of the latter is that any operator monotone function like x^{-1} or x^{1/2] has to be increasing, smooth, and concave.