Spectral norm of a matrix obtained by setting some entries to zero

Question

For example can we say, that if $A$ is original matrix and $A'$ obtained from $A$ by zeroing some elements then $\|A\|_2 \geq \|A'\|_2$?

Answer 1

Generally, this is not true. Consider $$ A=\begin{bmatrix} 1 & 1 \\ -1 & 1 \end{bmatrix}, \quad A'=\begin{bmatrix} 0 & 1 \\ -1 & 1 \end{bmatrix}. $$ Then $\|A\|_2=\sqrt{2}$, while $\|A'\|_2=(\sqrt{5}+1)/2>\|A\|_2$.

For non-negative matrices, we know that if $0\leq A'\leq A$ (in the component-wise sense), then $\rho(A')\leq\rho(A)$. So if $A$ and $A'$ are symmetric, we get the same inequality also for spectral norms. This holds also for nonsymmetric (nonnegative) $A$ and $A'$ as $0\leq A'\leq A$ implies that $0\leq A'^TA'\leq A^TA$. So $\|A'\|_2^2=\rho(A'^TA)\leq\rho(A^TA)=\|A\|_2^2$.