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Concept# Matrix norm

Summary

In mathematics, a matrix norm is a vector norm in a vector space whose elements (vectors) are matrices (of given dimensions).
Given a field of either real or complex numbers, let be the K-vector space of matrices with rows and columns and entries in the field . A matrix norm is a norm on .
This article will always write such norms with double vertical bars (like so: ). Thus, the matrix norm is a function that must satisfy the following properties:
For all scalars and matrices ,
(positive-valued)
(definite)
(absolutely homogeneous)
(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:
Every norm on Kn×n can be rescaled to be sub-multiplicative; in some books, the terminology matrix norm is reserved for sub-multiplicative norms.
Operator norm
Suppose a vector norm on and a vector norm on are given. Any matrix A induces a linear operator from to with respect to the standard basis, and one defines the corresponding induced norm or operator norm or subordinate norm on the space of all matrices as follows:
where denotes the supremum. This norm measures how much the mapping induced by can stretch vectors.
Depending on the vector norms , used, notation other than can be used for the operator norm.
If the p-norm for vectors () is used for both spaces and then the corresponding operator norm is:
These induced norms are different from the "entry-wise" p-norms and the Schatten p-norms for matrices treated below, which are also usually denoted by
In the special cases of the induced matrix norms can be computed or estimated by
which is simply the maximum absolute column sum of the matrix;
which is simply the maximum absolute row sum of the matrix.
For example, for
we have that
In the special case of (the Euclidean norm or -norm for vectors), the induced matrix norm is the spectral norm. (The two values do not coincide in infinite dimensions — see Spectral radius for further discussion.

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