Is the inequality true for matrices: $\|AB\|_{\infty} \leq\|A\|_{\infty}\|B\|_{\max}$?

Question

Is the inequality true for matrices: $\|AB\|_{\infty} \leq\|A\|_{\infty}\|B\|_{\max}$? Here A and B are both matrices having real-valued numbers.

Answer 1

Matrix norms are submultiplicative. That is, $\lVert A B \rVert \leq \lVert A \rVert \lVert B \rVert$.

Some examples of matrix norms are the sum of the absolute value of the entries, the frobenius norm, the norm induced by a vector norm (i.e. $\max_{\lVert x \rVert = 1 } \lVert A x \rVert $), etc.

Note that the maximum absolute value of all the entries in $A$ is not a matrix norm, but n*this is.

So, you have $\lVert A B \rVert_\infty \leq \lVert A \rVert_\infty \lVert B \rVert_\infty$ where $\lVert A \rVert_\infty $ is the maximum row sum of the absolute value of the entries.

Now, if by max, you mean the maximum of the absolute value of the entries, this isn't true. Take $A = I$, then you have to just check $\lVert B \rVert_\infty \leq \lVert B \rVert_{max}$. Take a matrix which has $1$ in the top left corner, zeros in the rest of the first row, and $ 2/n$ in the rest of the matrix. This would imply $2 < 1$ which is absurd.