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

$\backslash |AB\backslash |\backslash le\backslash |A\backslash |\; \backslash |B\backslash |$

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

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

$r\backslash left|A\backslash right|\_1\backslash leq\backslash left|A\backslash right|\_2\backslash leq\; s\backslash left|A\backslash right|\_1$

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

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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 maxima:

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

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

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

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

p-norms and uniform norms

For example, the operator norm corresponding to the p-norm for vectors is:

$\backslash left\; \backslash |\; A\; \backslash right\; \backslash |\; \_p\; =\; \backslash operatorname\{max\}\; \_\{x\; \backslash ne\; 0\}\; \backslash frac\{\backslash left\; \backslash |\; A\; x\backslash right\; \backslash |\; \_p\}\{\backslash left\; \backslash |\; x\backslash right\; \backslash |\; \_p\}$

In the case of $p=1$ and $p=\backslash infty$, the norms can be computed as:

$\backslash left\; \backslash |\; A\; \backslash right\; \backslash |\; \_1\; =\; \backslash operatorname\{max\}\; \_\{1\; \backslash leq\; j\; \backslash leq\; n\}\; \backslash sum\; \_\{i=1\}\; ^m\; |\; a\_\{ij\}\; |$

$\backslash left\; \backslash |\; A\; \backslash right\; \backslash |\; \_\backslash infty\; =\; \backslash operatorname\{max\}\; \_\{1\; \backslash leq\; i\; \backslash leq\; m\}\; \backslash sum\; \_\{j=1\}\; ^n\; |\; a\_\{ij\}\; |$

Spectral norm

In the special case of p = 2 (the Euclidean norm) and m = n (square matrices), 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^*:

$\backslash |A\backslash |=\backslash sqrt\{\backslash lambda\_\{max\}(A^*A)\}.$

Any induced norm satisfies the inequality

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

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

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

"Entrywise" norms

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These vector norms treat a matrix as an $m\; \backslash times\; n$ vector, and use one of the familiar vector norms. Most entrywise norms are not (submultiplicative) matrix norms.

For example, using the p-norm for vectors, we get:

$\backslash Vert\; A\; \backslash Vert\_\{p\}\; =\; \backslash Big(\; \backslash sum\_\{i=1\}^m\; \backslash sum\_\{j=1\}^n\; |a\_\{ij\}|^p\; \backslash Big)^\{1/p\}\; \backslash ,$

Frobenius norm

For p = 2, this is called the Frobenius norm. This norm can be defined in various ways:

$\backslash |A\backslash |\_F^2=\backslash sum\_\{i=1\}^m\backslash sum\_\{j=1\}^n\; |a\_\{ij\}|^2=\backslash operatorname\{trace\}(AA^H)=\backslash sum\_\{i=1\}^\{\backslash min\backslash \{m,\backslash ,n\backslash \}\}\; \backslash sigma\_\{i\}^2$

where A^H denotes the conjugate transpose of A, σ_i are the singular values of A, and the trace function is used. The Frobenius 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.

Consistent norms

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

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

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

References

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• Roger Horn and Charles Johnson. Matrix Analysis, Chapter 5, Cambridge University Press, 1985. ISBN 0-521-38632-2.
• L. Thomas, Norms and Condition Numbers of a Matrix ^*

Linear algebra