Matrix norm

For the general concept, see Norm (mathematics).

In mathematics, a matrix norm is a vector norm in a vector space whose elements (vectors) are matrices (of given dimensions).

Preliminaries

Given a field $K$ of either real or complex numbers, let $K^{m\times n}$ be the K-vector space of matrices with $m$ rows and $n$ columns and entries in the field $K$. A matrix norm is a norm on $K^{m\times n}$.

This article will always write such norms with double vertical bars (like so: $\|A\|$). Thus, the matrix norm is a function $\|\cdot \|:K^{m\times n}\to \mathbb {R}$ that must satisfy the following properties:

For all scalars $\alpha \in K$ and matrices $A,B\in K^{m\times n}$,

• $\|A\|\geq 0$ (positive-valued)
• $\|A\|=0\iff A=0_{m,n}$ (definite)
• $\left\|\alpha A\right\|=\left|\alpha \right|\left\|A\right\|$ (absolutely homogeneous)
• $\|A+B\|\leq \|A\|+\|B\|$ (sub-additive or satisfying the triangle inequality)

The only feature distinguishing matrices from rearranged vectors is multiplication. Matrix norms are particularly useful if they are also sub-multiplicative:

• $\left\|AB\right\|\leq \left\|A\right\|\left\|B\right\|$

Every norm on K^n×n can be rescaled to be sub-multiplicative; in some books, the terminology matrix norm is reserved for sub-multiplicative norms.

Matrix norms induced by vector norms

Main article: Operator norm

Suppose a vector norm $\|\cdot \|_{\alpha }$ on $K^{n}$ and a vector norm $\|\cdot \|_{\beta }$ on $K^{m}$ are given. Any $m\times n$ matrix A induces a linear operator from $K^{n}$ to $K^{m}$ with respect to the standard basis, and one defines the corresponding induced norm or operator norm or subordinate norm on the space $K^{m\times n}$ of all $m\times n$ matrices as follows:

$${\begin{aligned}\|A\|_{\alpha ,\beta }&=\sup\{\|Ax\|_{\beta }:x\in K^{n}{\text{ with }}\|x\|_{\alpha }=1\}\\&=\sup \left\{{\frac {\|Ax\|_{\beta }}{\|x\|_{\alpha }}}:x\in K^{n}{\text{ with }}x\neq 0\right\}.\end{aligned}}$$

where $\sup$ denotes the supremum. This norm measures how much the mapping induced by $A$ can stretch vectors. Depending on the vector norms $\|\cdot \|_{\alpha }$, $\|\cdot \|_{\beta }$ used, notation other than $\|\cdot \|_{\alpha ,\beta }$ can be used for the operator norm.

Matrix norms induced by vector p-norms

If the p-norm for vectors ($1\leq p\leq \infty$) is used for both spaces $K^{n}$ and $K^{m}$, then the corresponding operator norm is:

$${\Vert <!--\; \Vert \; -->A\Vert <!--\; \Vert \; -->}_p=\underset{x\ne <!--\; \ne \; -->0}{sup}\frac{{\Vert <!--\; \Vert \; -->Ax\Vert <!--\; \Vert \; -->}_p}{}$$