matrix p-norm

A class of matrix norms, denoted $\parallel \cdot {\parallel }_p$, is defined as

$${\parallel A\parallel }_p=\underset{x\ne 0}{sup}\frac{{\parallel Ax\parallel }_p}{{\parallel x\parallel }_p}\mathit{\hspace{1em}\hspace{1em}}x\in {ℝ}^n,A\in {ℝ}^{m\times n}.$$

The matrix $p$-norms are defined in terms of the vector $p$-norms (http://planetmath.org/VectorPNorm).

An alternate definition is

$${\parallel A\parallel }_p=\underset{{\parallel x\parallel }_p=1}{\max }{\parallel Ax\parallel }_p.$$

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

It directly follows from this that ${\parallel A\parallel }_1={\parallel A^T\parallel }_{\mathrm{\infty }}$.

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 ${\parallel A\parallel }_2$:

$$\frac{1}{\sqrt{n}}{\parallel A\parallel }_{\mathrm{\infty }}\le {\parallel A\parallel }_2\le \sqrt{m}{\parallel A\parallel }_{\mathrm{\infty }}.$$

$$\frac{1}{\sqrt{m}}{\parallel A\parallel }_1\le {\parallel A\parallel }_2\le \sqrt{n}{\parallel A\parallel }_1.$$

$${\parallel A\parallel }_2^2\le {\parallel A\parallel }_{\mathrm{\infty }}\cdot {\parallel A\parallel }_1.$$

$${\parallel A\parallel }_2\le {\parallel A\parallel }_F\le \sqrt{n}{\parallel A\parallel }_2.$$

(${\parallel A\parallel }_F$ is the Frobenius matrix norm)

Title	matrix p-norm
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