Why do we have the fact that $\|A-B\|_2\le d\epsilon_n \, ? $

Question

For two symmetric and semi-positive covariance matrices $A$ and $B$ of size $d\times d$, let $\epsilon_n=\max_{j,k}|A_{jk}-B_{jk}|. $ Why do we have the fact that $$ \|A-B\|_2\le d\epsilon_n \, ? $$

This means $$ \|A-B\|_2\le d\max_{j,k}|A_{jk}-B_{jk}| $$

Answer 1

Let $M = A - B$. Let $\|\cdot\|_F$ denote the Frobenius norm. Note that $$ \|M\|_2^2 \leq \|M\|_F^2 = \sum_{i,j} |M_{jk}|^2 \leq d^2 \max_{j,k} |M_{j,k}|^2 = d^2 \epsilon_n^2. $$