Vector and matrix norm definitions?

Question

I've questions on these four norms whose definitions I'm memorizing like this:

1. Vector euclidean norm: $(x_1^2+x_2^2+\cdots+x_n^2)^{1/2}$

2. Vector max norm: $\max\{|x_1|, |x_2|, \ldots, |x_n|\}$

3. Matrix norm: $\max\limits_{x \neq 0} \frac{\|Ax\|}{\|x\|}$

4. Matrix max norm: $\max\limits_{1<i<N} \sum\limits_{1<j<N}|a_{ij}|$

How to interpret these? The Vector euclidian norm is a scalar. The vector max norm is also a scalar. The matrix norm is also a scalar but the matrix max norm is a vector? Can you tell me more how to interpret these formulas? Are my formulae correct? Can they be more pedagogically written?

Answer 1

This got too long for a comment:

Using the definition of the vecto $p$ norm $$ \|\mathbf{x}\|_p := \bigg( \sum_{i=1}^n |x_i|^p \bigg)^{1/p}, $$ you can combine 3.) and 4.) like the following:

Let $$ \left \| A \right \| _p = \max \limits _{x \ne 0} \frac{\left \| A x\right \| _p}{\left \| x\right \| _p}. $$ So you get back 3.) with $p=2$. In the case of $p=1$ and $p=\infty$, the norms can be computed as:

• $ \left \| A \right \| _1 = \max \limits _{1 \leq j \leq n} \sum _{i=1} ^m | a_{ij} |, $ which is simply the maximum absolute column sum of the matrix.
• $ \left \| A \right \| _\infty = \max \limits _{1 \leq i \leq m} \sum _{j=1} ^n | a_{ij} |, $ which is simply the maximum absolute row sum of the matrix

Here I'm not sure, which of both you mean, but none of them is a vector as already pointed out in the comments (taken from Matrix norm/Induced_norm, which also provides some examples, that might help with the interpretation).