If $A \in \mathbb{R}^{n \times n}$ then $|a_{ij}| \leq ||A||.$

Question

Let $A \in \mathbb{R}^{n \times n}$ and $||\cdot||$ be an arbitrary matrx norm. I want to show that $|a_{ij}| \leq ||A||.$

I've been looking over the matrix norm properties, and I'm just not seeing a way to compare these two values. I've come across specific matrix norms, but this is an arbitrary one, and I don't know what to do with it.

Does anyone have a hint to point me in the right direction? Since I don't have any input myself, I'm not necessarily looking for a full answer.

Answer 1

This statement is not true. As an example, consider the matrix $$ A = \pmatrix{0 & 1\\0 & 0}, $$ and let $\|\cdot\|_k$ denote the matrix norm defined by $$ \|A\|_k = \|S_kAS_k^{-1}\|, \quad S = \pmatrix{k & 0\\ 0 & 1}, $$ where $\|A\|$ denotes the induced Euclidean norm (maximal singular value, AKA spectral norm) of $A$. We find that for every $k>0$, $\|\cdot \|_k$ is indeed a norm. However, $\|A\|_k = k$. So, if we select $k<1$, then we find that $|a_{12}| > \|A\|_k$, contradicting the claim.