TITLE: Proving that $\|A\|_{\infty}$ the largest row sum of absolute value of matrix $A$
QUESTION [11 upvotes]: I am studying matrix norms. I have read that $\|A\|_{\infty}$ is  the largest row sum of absolute value and $\|A\|_{1}$ is the highest column sum of absolute values of the matrix $A$. However, I am not able to prove this. Are there any proof of these statements? Please help  and thanks for your time.
Edit: Where $\|A\|_{\infty}$ is the matrix norm induced by the vector norm $\|x\|_{\infty}$.

REPLY [8 votes]: I assume that the author tries to derive the matrix norms $\|A\|_1$ and $\|A\|_\infty$ induced by vector norms $\|x\|_1$ and $\|x\|_\infty$.
Let's take, for example, $\|\cdot\|_\infty$. We write
$$\|Ax\|_\infty=\sup_i \left|\sum_j A_{ij}x_j\right|\le \sup_i  \sum_j |A_{ij}||x_j|\le \sup_j |x_j| \sup_i  \sum_j |A_{ij}|.$$
Hence, a good candidate for $\|A\|_\infty$ is $\sup_i  \sum_j |A_{ij}|$. We need to prove that this boundary is indeed achieved; it is true, since we can take $i$ where that supremum is achieved and impose $x_j=\mathrm{sign}(A_{ij})$. With such $x$ all our inequalities degenerate to equalities and we conclude that $\|A\|_\infty = \sup_i  \sum_j |A_{ij}|$ is a matrix norm induced by $\|x\|_\infty = \sup_j |x_j|$.
The case $\|\cdot\|_1$ is done likewise.