[Math] Matrix norm induced from vector explanation

Question

Can someone explain as thoroughly as possible what a matrix induced norm is? A concrete example would for this norm would help a lot…

$A \in C^{n \times n}, v \in C^n$

$|| A || = \displaystyle\sup_{v \neq 0}\frac{||Av||}{||v||}$

Thank you.

Answer 1

One can view the induced norm as the largest factor by which $A$ can stretch any vector (while also changing its direction).

In other words, if we denote $\varphi := \|A\|$, then $\|Av\| \le \varphi \|v\|$, for all $v$, and $\varphi$ is the smallest nonnegative real number with such a property.

Note that the "stretch" above means "make longer by a factor (shorter if that factor is less than 1)" and "length" is dependent of the choice of the vector norm.

This, of course, is not a formal definition, but more of a layman explanation if you have a hard time of grasping the concept.