matrix p-norm

A class of matrix normsMathworldPlanetmath, denoted p, is defined as

Ap=supx0Axpxp  xn,Am×n.

The matrix p-norms are defined in terms of the vector p-norms (

An alternate definition is


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


It directly follows from this that A1=AT.

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 eigenvalueMathworldPlanetmathPlanetmathPlanetmathPlanetmath of ATA. There are also various inequalitiesMathworldPlanetmath that allow one to make estimates on the value of A2:


(AF is the Frobenius matrix norm)

Title matrix p-normMathworldPlanetmath
Canonical name MatrixPnorm
Date of creation 2013-03-22 11:43:22
Last modified on 2013-03-22 11:43:22
Owner mathcam (2727)
Last modified by mathcam (2727)
Numerical id 20
Author mathcam (2727)
Entry type Definition
Classification msc 15A60
Classification msc 00A69
Related topic MatrixNorm
Related topic VectorNorm
Related topic FrobeniusMatrixNorm