# Vector and matrix norm definitions?

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?

• Please try to use TeX syntax for equations. I've done it for you, this time. Commented Aug 6, 2012 at 10:16
• The matrix norm is a scalar not a vector. The operator norm on matrices (for examples) is defined using vectors, but it is a scalar quantity -it is the maximum of a set of positive real numbers. Commented Aug 6, 2012 at 10:21
• Thank you for the comments. If all definitions are scalars then I've misunderstood and must go back and rehearse. Commented Aug 6, 2012 at 10:41
• For 1: it's geometrically the distance of a particular $n$-dimensional point from the origin of a Cartesian coordinate system. For 2: look up "chessboard distance". Commented Aug 6, 2012 at 10:46

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