Matrix norm

In mathematics, the term matrix norm can have two meanings:

1. A vector norm on matrices, i.e, a norm on the vector space of all real or complex m-by-n matrices.
2. A sub-multiplicative vector norm refers to a vector norm on square matrices compatible with matrix multiplication in the sense that

The set of all n-by-n matrices, together with such a sub-multiplicative norm, is a Banach algebra.

In the remaining article, we will follow the tradition in matrix theory. We use term "vector norm" for the first definition and "matrix norm" for the second definition.

Table of contents

1 Equivalent norm 2 Operator norm or Induced norm 3 Spectral norm or Spectral radius 4 Frobenius norm

Equivalent norm

For any two vector norm | · | and | · |₁, we have

r|A|₁≤ |A|₂≤ s|A|₁

for some positive number r and s, for all matrices A. In order words, they are equivalent norms.

Moreover, when m=n, then for any vector norm | · |, there exists a unique positive number k such that k| · | is a (submultiplicative) matrix norm.

A matrix norm || · || is said to be minimal if there exists no other matrix norm | · | satisfying |A|≤||A|| for all |A|.

Operator norm or Induced norm

If norms on K^m and Kⁿ are given (K is real or complex), then one defines the corresponding induced norm or operator norm on the space of m-by-n matrices as the following suprema:

If m = n and one uses the same norm on domain and range, then these operator norms are all (submultiplicative) matrix norms.

Spectral norm or Spectral radius

If m=n and the norm on Kⁿ is the Euclidean norm, then the induced matrix norm is the spectral norm.

Spectral norm is the only minimal matrix norm which is an induced norm. The spectral norm of A equals to the square root of the spectral radius of AA_* or the largest singular value of A.

An important property for matrix norm is

where ρ(A) is the spectral radius of A.

Frobenius norm

The Frobenius norm of A is defined as

where A^* denotes the conjugate transpose of A, σ_i are the singular values 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.