# 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 (http://planetmath.org/VectorPNorm).

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)

Title matrix p-norm MatrixPnorm 2013-03-22 11:43:22 2013-03-22 11:43:22 mathcam (2727) mathcam (2727) 20 mathcam (2727) Definition msc 15A60 msc 00A69 MatrixNorm VectorNorm FrobeniusMatrixNorm