Matrix norm induced from vector explanation 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.
 A: 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.
A: Hints: $Av$ is another vector when $v$ is a vector. Assuming one puts the Euclidean norm on $\mathbb{C}^{n},$ the norm of a vector $u \in \mathbb{C}^{n}$ is given by $\| u \| = \sqrt{\sum_{i=1}^{n} |u_{i}|^{2}}$, where the $u_{i}$ are the entries of $u$ (though other norms are possible). Armed with this information, to calculate the norm of a matrix is a matter of following the definitions. The sup is the least upper bound. It may help to note 
that the ratio $\frac{\|Av\|}{\|v\|}$ is unchanged if we multiply $v$ by a non-zero scalar, so it suffices to consider vectors with $\| v \| = 1.$
