# Submultiplicity of Induced Norms

This is a lecture note on induced norms from Cornell: pdf

Given $A$ is a matrix and $v$ is a vector with length less than or equal to $1$. It says, "if $||\cdot||$ is an induced norm, then $||Av||\leq ||A||\cdot||v||$ from the definition of vector norms."

I check all definitions on Wikipedia, but could not figure out why. Is there any hint or comment? I don't think vector norms have such properties.

Hint:

Let me write your inequality when $v \ne 0$ as

$$\frac{\|Av\|}{\|v\|}\le \|A\|$$

• If I change "$||v||=1$" to "$||v||\leq1$" in the definition of induced norms, will we still have such inequality?
– user539442
Commented Apr 16, 2018 at 23:48
• maximizing $\|Av\|$ over $\{v: \|v\|=1\}$ is equal to maximizing $\|Av\|$ over $\{v: \|v\|\le 1\}$. Use the property that $\|cv\|=|c|\|v\|$ to prove it. Commented Apr 16, 2018 at 23:52
• hmm...the question used $\|.\|$ for both vector norm and induced norm. Commented Apr 17, 2018 at 3:11

Nah, I think they just mean 'from the definition of induced norm'. Since the norm of $A$ is defined as the supremum, we have inequality for all $v \in V$.