Matrix Norm Definition

Question

I don't understand the intuition behind the definition of a matrix norm: $\displaystyle \|A\|_p = \max_{x\neq0}\frac{\|Ax\|_p}{\|x\|}$.

Why is the arbitrary vector $\vec{x}$ included in the expression? Intuitively, what does this mean?

Answer 1

One way to think of the matrix norm is a way of measuring the deformation of the unit sphere under transformation by $A$ - another way to write $\|A\|_p$ is $$\max_{\|x\| = 1} \|Ax\|_p.$$ It is measuring the largest $p$ norm of $Ax$ among all unit vectors $x$.

Answer 2

To say that the norm is $N$ means the mapping cannot take any vector to a vector that's more than $N$ times as big, although it can take some vectors to a vector that is exactly $N$ times as big.