In mathematics, the term matrix norm can have two meanings:
- A vector norm on matrices, i.e, a norm on the vector space of all real or complex m-by-n matrices.
- A sub-multiplicative vector norm refers to a vector norm on square matrices compatible with matrix multiplication in the sense that
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.
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2 Operator norm or Induced norm 3 Spectral norm or Spectral radius 4 Frobenius norm |
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|.
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
Equivalent norm
For any two vector norm | · | and | · |1, we have
for some positive number r and s, for all matrices A. In order words, they are equivalent norms.Operator norm or Induced norm
If norms on Km 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.
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, σi 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.
