(2.10) — $\|\sum_j u_jv_j^\top\|_\infty=1$ and $\langle\sum_j u_jv_j^\top,B\rangle=\langle\sum_j u_jv_j^\top,P_{S_1}BP_{S_2}\rangle$

Let $A=\sum_{j=1}^r\sigma_ju_jv_j^\top$ be a singular value decomposition of a real $m_1\times m_2$ matrix, with support $(S_1,S_2)$ and $P_{S_1}$, $P_{S_2}$ the orthogonal projectors onto $S_1$, $S_2$. Then

1. if $A\ne0$ (that is, $r\ge1$), $\big\|\sum_{j=1}^ru_jv_j^\top\big\|_\infty=1$ (operator norm);
2. for every $m_1\times m_2$ matrix $B$,

In the proof of Theorem 1 these are applied with $B=\hat A^\lambda-A$ to bound the term $-\lambda\langle\sum_ju_jv_j^\top,\hat A^\lambda-A\rangle$ of (2.8).

Formalization Note The paper states the first fact without restriction; for $A=0$ the sum is empty and its norm is $0$, so the condition $r>0$ is added (it is the case the proof uses: for $A=0$ the term vanishes). The paper writes $B=\hat A^\lambda-A$; the second fact holds for every $B$ and is stated so.