Matrix norm

From Exampleproblems

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 is any vector norm on square matrices compatible with matrix multiplication in the sense that

$\|AB\|\le\|A\| \|B\|$

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

In the rest of the 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.

Equivalence of norms

For any two vector norms | · |₁ and | · |₂, we have

for some positive numbers r and s, for all matrices A. In other words, they are equivalent norms; they induce the same topology on the real or complex vector space.

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:

$\|A\|=\sup\{\|Ax\| : x\in K^n \mbox{ with }\|x\|\le 1\}$

$= \sup\{\|Ax\| : x\in K^n \mbox{ with }\|x\| = 1\}$

$= \sup\left\{\frac{\|Ax\|}{\|x\|} : x\in K^n \mbox{ with }x\ne 0\right\}$

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

Consistent norms

A matrix norm $\| \cdot \|_{ab}$ on $K^{m \times n}$ is called consistent with a vector norm $\| \cdot \|_{a}$ on $K n$ and a vector norm $\| \cdot \|_{b}$ on $K m$ if:

$\|Ax\|_b \leq \|A\|_{ab} \|x\|_a$

for all $A \in K^{m \times n}, x \in K^n$ . All induced norms are consistent by definition.

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. The spectral norm of a matrix A is the largest singular value of A or the square root of the largest eigenvalue of the positive-semidefinite matrix AA^*:

$\|A\|=\sqrt{\lambda_{max}(A^*A)}.$

Any induced norm satisfies the inequality

$\|A\| \ge \rho(A),$

where ρ(A) is the spectral radius of A. Furthermore, we have

$\lim_{r\rarr\infty}\|A^r\|^{1/r}=\rho(A).$

"Entrywise" norms

These vector norms treat a matrix as an $m \times n$ vector, and use one of the familiar vector norms. For example, for k=1,2,..., we have the following k-norm:

$\Vert A \Vert_{k} = \{\sum_{m,n} |a_{mn}|^k\}^{1/k}$

For k=2, it corresponds to the Euclidean norm and is called the Frobenius norm. Most entrywise norms are not (submultiplicative) matrix norms.

Frobenius norm

The Frobenius norm of A is defined as

$\|A\|_F^2=\sum_{i=1}^m\sum_{j=1}^n |a_{ij}|^2=\operatorname{trace}(AA^*)=\sum_{i=1}^{\min\{m,\,n\}} \sigma_{i}^2$

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.

The Frobenius norm is submultiplicative and is very useful for numerical linear algebra. This norm is often more natural and more convenient than the induced norms.

References

Roger Horn and Charles Johnson. Matrix Analysis, Chapter 5, Cambridge University Press, 1985. ISBN 0-521-38632-2.

Matrix norm

From Exampleproblems

Contents

Equivalence of norms

Operator norm or induced norm

Consistent norms

Spectral norm or spectral radius

"Entrywise" norms

Frobenius norm

References

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