How do you show that the norm for square matrices is submultiplicative?

Question

In these notes: http://www.math.usm.edu/lambers/mat610/sum10/lecture2.pdf

It says that square matrices satisfy so called submultiplicative norm $\|AB\| \leq \|A\|\|B\|$. Is it by definition or is there some way to prove this?

Answer 1

Not all norms that can be given to matrices have this property. However, there are a family of important matrix norms, called "induced norms" (also "operator norms"), which are defined by

$$\| A \| = \sup_{\| x \| = 1} \| A x \|$$

for some vector norm $\| \cdot \|$. Such norms are indeed submultiplicative, because if $\| x \| = 1$ then $\| AB x \| \leq \| A \| \| Bx \| \leq \| A \| \| B \| \| x \| = \| A \| \| B \|$.

Answer 2

It is not always the case. There are matrix norms that do not have this property. That depends on that definition of the matrix norm we are using, since there are many. To name a few:

1) Operator norm (induced norms)

2) Frobenius norm

3) P norm