Is the following result true $\|Ax - Ay\|_p \leq \|A\|_p\|x-y\|_p$

Question

Is the following true:

Claim: Let $x,y \in \mathcal{D} \subset \mathbb{R}^n$, for $1\leq p \leq \infty$, $A \in \mathbb{R}^{n \times n}$

$$\|Ax - Ay\|_p \leq \|A\|_p\|x-y\|_p$$

I am not that familiar with matrix norms, but according to here: http://www.math.drexel.edu/~foucart/TeachingFiles/F12/M504Lect6.pdf

$\|A\|_p = \max\limits_{\|x\|_p = 1} \|Ax\|_p$

Does anyone know if the claim is true? and how do you supposed to show it?

Answer 1

Hint: $$\|A x - A y\|_p = \|x - y\|_p \left\|A \left( \frac{x-y}{\|x-y\|_p}\right)\right\|_p$$

Answer 2

$||Ax-Ay||_p=||A(x-y)||_p\le ||A||_p ||x-y||_p$

where $||A||=\sup_{x\neq 0} \frac{||Ax||}{||x||}$