# Matrix norm

In mathematics, a matrix norm is a natural extension of the notion of a vector norm to matrices.

## Definition

Additionally, in the case of square matrices (thus, m = n), some (but not all) matrix norms satisfy the following condition, which is related to the fact that matrices are more than just vectors:

A matrix norm that satisfies this additional property is called a sub-multiplicative norm (in some books, the terminology matrix norm is used only for those norms which are sub-multiplicative). The set of all n-by-n matrices, together with such a sub-multiplicative norm, is an example of a Banach algebra.

## Induced norm

{{#invoke:main|main}} If vector norms on Km and Kn are given (K is field of real or complex numbers), then one defines the corresponding induced norm or operator norm on the space of m-by-n matrices as the following maxima:

{\begin{aligned}\|A\|&=\sup\{\|Ax\|:x\in K^{n}{\mbox{ with }}\|x\|=1\}\\&=\sup \left\{{\frac {\|Ax\|}{\|x\|}}:x\in K^{n}{\mbox{ with }}x\neq 0\right\}.\end{aligned}} The operator norm corresponding to the p-norm for vectors is:

$\left\|A\right\|_{p}=\sup \limits _{x\neq 0}{\frac {\left\|Ax\right\|_{p}}{\left\|x\right\|_{p}}}.$ These are different from the entrywise p-norms and the Schatten p-norms for matrices treated below, which are also usually denoted by $\left\|A\right\|_{p}.$ $\left\|A\right\|_{1}=\max \limits _{1\leq j\leq n}\sum _{i=1}^{m}|a_{ij}|,$ which is simply the maximum absolute column sum of the matrix.
$\left\|A\right\|_{\infty }=\max \limits _{1\leq i\leq m}\sum _{j=1}^{n}|a_{ij}|,$ which is simply the maximum absolute row sum of the matrix

For example, if the matrix A is defined by

$A={\begin{bmatrix}-3&5&7\\2&6&4\\0&2&8\\\end{bmatrix}},$ then we have ||A||1 = max(|-3|+2+0, 5+6+2, 7+4+8) = max(5,13,19) = 19. and ||A|| = max(|-3|+5+7, 2+6+4,0+2+8) = max(15,12,10) = 15.

{{safesubst:#invoke:anchor|main}}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 i.e. the square root of the largest eigenvalue of the positive-semidefinite matrix A*A:

$\left\|A\right\|_{2}={\sqrt {\lambda _{\text{max}}(A^{^{*}}A)}}=\sigma _{\text{max}}(A)$ where A* denotes the conjugate transpose of A.

$\left\|A\right\|_{\alpha ,\beta }=\max \limits _{x\neq 0}{\frac {\left\|Ax\right\|_{\beta }}{\left\|x\right\|_{\alpha }}}.$ Subordinate norms are consistent with the norms that induce them, giving

$\|Ax\|_{\beta }\leq \|A\|_{\alpha ,\beta }\|x\|_{\alpha }.$ Any induced norm satisfies the inequality

$\left\|A^{r}\right\|^{1/r}\geq \rho (A),$ where ρ(A) is the spectral radius of A. For a symmetric or hermitian matrix $A$ , we have equality for the 2-norm, since in this case the 2-norm is the spectral radius of $A$ . For an arbitrary matrix, we may not have equality for any norm. Take

$A={\begin{bmatrix}0&1\\0&0\\\end{bmatrix}},$ the spectral radius of $A$ is 0, but $A$ is not the zero matrix, and so none of the induced norms are equal to the spectral radius of $A$ .

Furthermore, for square matrices we have the spectral radius formula:

$\lim _{r\rightarrow \infty }\|A^{r}\|^{1/r}=\rho (A).$ ## "Entrywise" norms

These vector norms treat an $m\times n$ matrix as a vector of size $mn$ , and use one of the familiar vector norms.

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

$\Vert A\Vert _{p}=\Vert {\mathrm {vec} }(A)\Vert _{p}=\left(\sum _{i=1}^{m}\sum _{j=1}^{n}|a_{ij}|^{p}\right)^{1/p}$ This is a different norm from the induced p-norm (see above) and the Schatten p-norm (see below), but the notation is the same.

The special case p = 2 is the Frobenius norm, and p = ∞ yields the maximum norm.

### L2,1 norm

$\Vert A\Vert _{2,1}=\sum _{j=1}^{n}\Vert a_{j}\Vert _{2}=\sum _{j=1}^{n}\left(\sum _{i=1}^{m}|a_{ij}|^{2}\right)^{1/2}$ Note here the two indexes $i,j$ of $A_{i,j}$ are treated differently; all matrix norms introduced prior to the L2,1 norm treat the two indexes symmetrically. L2,1 norm is widely used in robust data analysis and sparse coding for feature selection.

$\Vert A\Vert _{p,q}=\left[\sum _{j=1}^{n}\left(\sum _{i=1}^{m}|a_{ij}|^{p}\right)^{q/p}\right]^{1/q}$ ### Frobenius norm

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For p = q = 2, this is called the Frobenius norm or the Hilbert–Schmidt norm, though the latter term is often reserved for operators on Hilbert space. This norm can be defined in various ways:

$\|A\|_{F}={\sqrt {\sum _{i=1}^{m}\sum _{j=1}^{n}|a_{ij}|^{2}}}={\sqrt {\operatorname {trace} (A^{{}^{*}}A)}}={\sqrt {\sum _{i=1}^{\min\{m,\,n\}}\sigma _{i}^{2}}}$ where A* denotes the conjugate transpose of A, σi are the singular values of A, and the trace function is used. The Frobenius norm is similar to the Euclidean norm on Kn and comes from the Frobenius inner product on the space of all matrices.

The Frobenius norm is sub-multiplicative and is very useful for numerical linear algebra. This norm is often easier to compute than induced norms and has the useful property of being invariant under rotations. This property follows easily from the trace definition restricted to real matrices,

$\|A\|_{F}^{2}=\|P^{\top }\cdot B\cdot P\|_{F}^{2}=\operatorname {trace} \left(\left(P^{\top }\cdot B\cdot P\right)^{\top }\cdot \left(P^{\top }\cdot B\cdot P\right)\right)=\operatorname {trace} (B^{\top }\cdot B)=\|B\|_{F}^{2}$ ,

where we have used the orthogonal nature of P, $P^{\top }\cdot P={\mathbf {I} }$ and the cyclic nature of the trace, $\operatorname {trace} (XYZ)=\operatorname {trace} (ZXY)$ . More generally the norm is invariant under a unitary transformation for complex matrices.

### Max norm

The max norm is the elementwise norm with p = ∞:

$\|A\|_{\text{max}}=\max\{|a_{ij}|\}.$ This norm is not sub-multiplicative.

## Schatten norms

The Schatten p-norms arise when applying the p-norm to the vector of singular values of a matrix. If the singular values are denoted by σi, then the Schatten p-norm is defined by

$\|A\|_{p}=\left(\sum _{i=1}^{\min\{m,\,n\}}\sigma _{i}^{p}\right)^{1/p}.\,$ These norms again share the notation with the induced and entrywise p-norms, but they are different.

All Schatten norms are sub-multiplicative. They are also unitarily invariant, which means that ||A|| = ||UAV|| for all matrices A and all unitary matrices U and V.

The most familiar cases are p = 1, 2, ∞. The case p = 2 yields the Frobenius norm, introduced before. The case p = ∞ yields the spectral norm, which is the matrix norm induced by the vector 2-norm (see above). Finally, p = 1 yields the nuclear norm (also known as the trace norm, or the Ky Fan 'n'-norm), defined as

$\|A\|_{*}=\operatorname {trace} \left({\sqrt {A^{*}A}}\right)=\sum _{i=1}^{\min\{m,\,n\}}\sigma _{i}.$ ## Consistent norms

$\|Ax\|_{b}\leq \|A\|_{ab}\|x\|_{a}$ for all $A\in K^{m\times n},x\in K^{n}$ . All induced norms are consistent by definition.

## Compatible norms

$\|Ax\|_{a}\leq \|A\|_{b}\|x\|_{a}$ for all $A\in K^{n\times n},x\in K^{n}$ . Induced norms are compatible by definition.

## Equivalence of norms

$r\left\|A\right\|_{\alpha }\leq \left\|A\right\|_{\beta }\leq s\left\|A\right\|_{\alpha }$ ### Examples of norm equivalence

Here, $\|A\|_{p}$ refers to the matrix norm induced by the vector p-norm.

Another useful inequality between matrix norms is

$\|A\|_{2}\leq {\sqrt {\|A\|_{1}\|A\|_{\infty }}}.$ 