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I've been given this problem:

Prove that a subordinate matrix norm is a matrix norm, i.e.

if $\left \|. \right \|$ is a vector norm on $\mathbb{R}^{n}$, then $\left \| A \right \|=\max_{\left \| x \right \|=1}\left \| Ax \right \|$ is a matrix norm

I don't even understand the question, and a explanation on what the problem ask me to do would be very appreciated, thanks in advance.

specific what does $\max_{\left \| x \right \|=1}\left \| Ax \right \|$ mean

2 Answers 2

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A mapping $\|.\|$ from $\mathbb R^{n\times n}$ to $\mathbb [0,\infty)$ is a norm if, for all matrices $A, B$ and all scalars $\alpha$:

  • $\|A\|=0$ if and only if $A=0$
  • $\|\alpha A\| = |\alpha|\|A\|$
  • $\|A+B\|\leq \|A\|+\|B\|$
  • $\|AB\|\leq \|A\|\|B\|$

So the task is asking of you to verify all those properties are true in your case.

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    but what the notation $max_{\left \| x \right \|=1}\left \| Ax \right \|$ mean? – 2017-01-30
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    @stenvikteam it means $$\sup_{\left \| x \right \|=1}\left \| Ax \right \|$$, do you know what $\sup$ means? (Btw the reason why we write $\max$ for $\sup$ here is because the unit sphere in $\Bbb R^n$ is *compact*, which would not be true in infinite-dimensional spaces) – 2017-01-30
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    @Vim Yes I do. thank you very much. – 2017-01-30
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A Matrix Norm has to fullfill certain requirements. First, is has to be a norm, so it has to fullfill the three basic norm-Requirements.

1) Positivity

2) Linear wrt. to a constant

3) Triangular inequality

These are quite easy to show here.

Some Sources need a fourth thing, submultiplicativity. It means $\|A\| \cdot \|B\| \geq \|AB\|$ for all $A$, $B$. This, again, is easy to show, when you substitute your definition.

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    2) is commonly referred to as (positive) homogeneity. – 2017-01-30