Bound on the $2$-norm of a matrix with a single entry replaced by $0$

Question

Let $A \in \mathbb{R}^{n \times n}$ be a real matrix and $A'$ any matrix obtained from $A$ by replacing a single entry by $0$. It seems to hold experimentally that $\lVert A' \rVert_2 \leq \lVert A \rVert_2$, but I can't find a proof after a bunch of attempts. Any ideas?

Answer 1

This isn't true in general. Counterexample: $$ \left\|\pmatrix{1&1\\ 1&-1}\right\|=\sqrt{2}=1.4142 <1.6180=\frac{1+\sqrt{5}}{2}=\left\|\pmatrix{1&1\\ 1&0}\right\|. $$ However, if $A$ is entrywise nonnegative, it is true that $\|B\|_2\le\|A\|_2$ when $0\le B\le A$ entrywise. This is because $0\le B^TB\le A^TA$ implies that $\rho(B^TB)\le\rho(A^TA)$ (which follows from the more general fact that $X\le Y\Rightarrow\rho(X)\le\rho(Y)$ for nonnegative matrices $X$ and $Y$).