A
matrix norm is a
norm on the
vector space of all
real or
complex m-by-
n matrices. These norms are used to measure the "sizes" of matrices, and allow to talk about
limits of
sequences and
infinite series of matrices. Several different matrix norms ||.|| are in common use. The more important ones in the case
m =
n are compatible with
matrix multiplication in the sense that
- <math>\|AB\|\le\|A\| \|B\|</math>
The set of all
n-by-
n matrices, together with such a sub-multiplicative norm, is a
Banach algebra.
Suppose A=(aij) is an m-by-n matrix with entries from the field K (which is either R or C). The Frobenius norm of A is defined as
- <math>\|A\|_F=\sqrt{\sum_{i=1}^m\sum_{j=1}^n |a_{ij}|^2}=\operatorname{trace}(AA^*)</math>
where
A* denotes the
conjugate transpose of
A and the
trace function is used. This norm is very similar to the Euclidean norm on
Kn and comes from an
inner product on the space of all matrices; however, it is not sub-multiplicative for
m=
n.
If norms on Km and Kn are given, then one defines the corresponding operator norm[?] on the space of m-by-n matrices as the following suprema:
- <math>\|A\|=\sup\{\|Ax\| : x\in K^n \mbox{ with }\|x\|\le 1\}</math>
- <math>= \sup\{\|Ax\| : x\in K^n \mbox{ with }\|x\| = 1\}</math>
- <math>= \sup\left\{\frac{\|Ax\|}{\|x\|} : x\in K^n \mbox{ with }x\ne 0\right\}</math>
If
m =
n and one uses the same norm on domain and range, then these operator norms are all sub-multiplicative and give rise to Banach algebras.
The most "natural" of these operator norms is the one which arises from the Euclidean norms ||.||2 on
Km and Kn. It is unfortunately relatively difficult to compute; we have
- <math>\|A\|_2=\mbox{ the largest singular value of } A</math>
(see
singular value). If we use the taxicab norm ||.||
1 on
Km and
Kn, then we obtain the operator norm
- <math>\|A\|_1=\max_{1\le j\le n} \sum_{i=1}^m |a_{ij}|</math>
and if we use the maximum norm ||.||
∞ on
Km and
Kn, we get
- <math>\|A\|_\infty=\max_{1\le i\le m} \sum_{j=1}^n |a_{ij}|</math>
The following inequalities obtain among the various discussed matrix norms for the
m-by-
n matrix
A:
- <math>
\frac{1}{\sqrt{n}}\Vert\,A\,\Vert_\infty \leq \Vert\,A\,\Vert_2 \leq \sqrt{m}\Vert\,A\,\Vert_\infty
</math>
- <math>
\frac{1}{\sqrt{m}}\Vert\,A\,\Vert_1 \leq \Vert\,A\,\Vert_2 \leq \sqrt{n}\Vert\,A\,\Vert_1
</math>
- <math>
\Vert\,A\,\Vert_2 \leq \Vert\,A\,\Vert_F\leq\sqrt{n}\Vert\,A\,\Vert_2
</math>