PlanetMath: matrix p-norm

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matrix p-norm

(Definition)

A class of matrix norms, denoted $\Vert\cdot\Vert_p$ , is defined as

$\displaystyle \Vert\,A\,\Vert_p = \sup_{x\neq0}\frac{\Vert\,Ax\,\Vert_p}{\Vert\,x\,\Vert_p} \qquad{}x\in\mathbb{R}^n,A\in\mathbb{R}^{m\times n}.$

The matrix -norms are defined in terms of the vector -norms.

An alternate definition is

$\displaystyle \Vert\,A\,\Vert_p = \max_{\Vert\,x\,\Vert_p=1}\Vert\,Ax\,\Vert_p.$

As with vector -norms, the most important are the 1, 2, and $\infty$ norms. The 1 and $\infty$ norms are very easy to calculate for an arbitrary matrix:

$\begin{displaymath} \begin{array}{ll} \Vert\,A\,\Vert_1 & = \displaystyle\max_{1... ...e\max_{1\leq i\leq m}\sum_{j=1}^n\vert a_{ij}\vert. \end{array}\end{displaymath}$

It directly follows from this that $\Vert\,A\,\Vert_1 = \Vert\,A^T\,\Vert_\infty$ .

The calculation of the -norm is more complicated. However, it can be shown that the 2-norm of is the square root of the largest eigenvalue of . There are also various inequalities that allow one to make estimates on the value of $\Vert\,A\,\Vert_2$ :

$\displaystyle \frac{1}{\sqrt{n}}\Vert\,A\,\Vert_\infty \leq \Vert\,A\,\Vert_2 \leq \sqrt{m}\Vert\,A\,\Vert_\infty.$

$\displaystyle \frac{1}{\sqrt{m}}\Vert\,A\,\Vert_1 \leq \Vert\,A\,\Vert_2 \leq \sqrt{n}\Vert\,A\,\Vert_1 .$

$\displaystyle \Vert\,A\,\Vert_2^2\leq\Vert\,A\,\Vert_\infty\cdot\Vert\,A\,\Vert_1.$

$\displaystyle \Vert\,A\,\Vert_2 \leq \Vert\,A\,\Vert_F\leq\sqrt{n}\Vert\,A\,\Vert_2.$

( $\Vert\,A\,\Vert_F$ is the Frobenius matrix norm)

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Keywords:

frobenius

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Cross-references: Frobenius matrix norm, estimates, inequalities, eigenvalue, square root, calculate, norms, vector, terms, matrix, matrix norms, class
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This is version 10 of matrix p-norm, born on 2001-10-06, modified 2006-01-24.
Object id is 108, canonical name is MatrixPnorm.
Accessed 20551 times total.

Classification:

AMS MSC:

15A60 (Linear and multilinear algebra; matrix theory :: Norms of matrices, numerical range, applications of functional analysis to matrix theory)

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