Vector and Matrix norms

For any vector $\vv\in\R^n$ its length \[ \|\vv\| = \sqrt{v_1^2 + \cdots + v_n^2} \] is a measure for the “size” of the vector. It is however not always the most convenient or most relevant measure, and many alternatives have been introduced. One of these is the sum norm, \[ \|\vv\|_{1} = |v_1| + \cdots + |v_n|. \] Another is the maximum- or max norm \[ \|\vv\|_\max = \max\bigl\{ |v_1|, \dots, |v_n|\bigr\}. \]

Matrix norm

There are many ways of measuring “how large a matrix is.” One convenient measure of “the size of a matrix” is its largest row sum.

Definition. The largest row sum of a matrix

$\displaystyle A = \begin{pmatrix} a_{11} & a_{12} & \dots & a_{1n} \\ a_{21} & a_{22} & \dots & a_{2n} \\ \vdots& &\ddots& \\ a_{n1} & a_{n2} & \dots & a_{nn} \end{pmatrix} $

$\displaystyle \|A\| = \max_{1\le i\le n} \sum_{i=1}^n |a_{ij}| $

The maximal row sum of a matrix is compatible with the max norm of vectors, in the sense that the following holds

Theorem. For any vector $\vv\in\R^n$, and any $m\times n$ matrix $A$ we have \[ \|A\vv \|_\max \le \|A\|\, \|\vv\|_\max\,. \]

Proof

Let $1\le j\le m$ be given. Then \[ (A\vv)_j = a_{j1}v_1 + \cdots + a_{jn}v_n, \] so that \begin{align*} \left|(A\vv)_j\right| &\leq |a_{j1}| \, |v_1| + \cdots + |a_{jn}|\, |v_n| \\ &\leq \bigl\{|a_{j1}| + \cdots + |a_{jn}|\bigr\} \max\{|v_1|, \dots, |v_n|\} \\ &= \bigl\{|a_{j1}| + \cdots + |a_{jn}|\bigr\} \|\vv\|_\max \\ &\leq \|A\|\,\|\vv\|_\max . \end{align*}

Norm Properties

\[ \|A + B\| \leq \|A\| + \|B\| \qquad \forall A,B \] \[ \|tA\| = |t|\, \|A\| \qquad \forall t\in\R \] \[ \|AB\| \leq \|A\|\; \|B\| \qquad \forall A,B \] \[ \forall i,j : \left|a_{ij}\right| \le \|A\|. \]