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Homematrix p-norm

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# matrix p-norm

A class of matrix norms, denoted $\|\cdot\|_{p}$, is defined as

$\|\,A\,\|_{p}=\sup_{{x\neq 0}}\frac{\|\,Ax\,\|_{p}}{\|\,x\,\|_{p}}\qquad{}x\in% \mathbb{R}^{n},A\in\mathbb{R}^{{m\times n}}.$ |

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

An alternate definition is

$\|\,A\,\|_{p}=\max_{{\|\,x\,\|_{p}=1}}\|\,Ax\,\|_{p}.$ |

As with vector $p$-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:

$\begin{array}[]{ll}\|\,A\,\|_{1}&=\displaystyle\max_{{1\leq j\leq n}}\sum_{{i=% 1}}^{m}|a_{{ij}}|\\ \|\,A\,\|_{\infty}&=\displaystyle\max_{{1\leq i\leq m}}\sum_{{j=1}}^{n}|a_{{ij% }}|.\end{array}$ |

It directly follows from this that $\|\,A\,\|_{1}=\|\,A^{T}\,\|_{\infty}$.

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

$\frac{1}{\sqrt{n}}\|\,A\,\|_{\infty}\leq\|\,A\,\|_{2}\leq\sqrt{m}\|\,A\,\|_{% \infty}.$ |

$\frac{1}{\sqrt{m}}\|\,A\,\|_{1}\leq\|\,A\,\|_{2}\leq\sqrt{n}\|\,A\,\|_{1}.$ |

$\|\,A\,\|_{2}^{2}\leq\|\,A\,\|_{\infty}\cdot\|\,A\,\|_{1}.$ |

$\|\,A\,\|_{2}\leq\|\,A\,\|_{F}\leq\sqrt{n}\|\,A\,\|_{2}.$ |

($\|\,A\,\|_{F}$ is the *Frobenius matrix norm*)

Keywords:

frobenius

Related:

MatrixNorm, VectorNorm, FrobeniusMatrixNorm

Type of Math Object:

Definition

Major Section:

Reference

Groups audience:

## Mathematics Subject Classification

15A60*no label found*00A69

*no label found*

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## Comments

## the infinity-norm

I think the indexes are wrong in the definition of the infinity-norm.

Am I right?

D.A.

## Re: the infinity-norm

I think the entry is correct as it stands: the infinity norm is the largest row sum. Swapping the indices would make it equal to the 1 norm.

Lachlan

## Re: the infinity-norm

To Whom It May Concern,

I have to agree with Lachlan. You would not swap the indices, you would swap their upper bounds. The sum should be over n, and i should be bounded thusly: 1<=i<=m. Otherwise, the sum will not account for all the elements of the ith row in the event that m<n, and in the evaluation of the maximum, all the rows will not be accounted for in the event that n<m.

James

## bounds need correction

In A_\infinity norm , the bounds need correction, as has already been pointed some other users. Thanks.