Upper bound on $L_\infty$ norm of product of matrices

Question

Let $M_1$ and $M_2$ be two $n \times n$ matrices. Suppose, $||M_1||_\infty \leq U_1$ and $||M_2||_\infty \leq U_2$. What is the upper bound on $||M_1.M_2||_\infty$? Here is my analysis. $||M_1.M_2||_\infty = \max_{x} \frac{ ||M_1 . M_2 . x||_\infty }{||X||_\infty} = \max_{x} \frac{ ||M_1 . M_2 . x||_\infty . ||M_2 . x||_\infty }{||M_2.x||_\infty .||x||_\infty} \leq ||M_1||_\infty . ||M_2||_\infty$

Is my analysis correct? Does the same analysis work even for other matrix norms such as $L_n$ for any integer $n$?

Answer 1

This is correct.

Note that $\|M\|_\infty = \max_i \sum_j |m_{ij}|$.

The norm $\|\cdot \|_\infty$ is submultiplicative on the ring of $n \times n$ matrices. The inequality $\|M_1M_2\|_\infty \le \|M_1\|_\infty \|M_2\|_\infty $ is sharp if e.g. all entries of each matrix are the same constant.