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In the field ofmathematics,norms are defined for elements within avector space. Specifically, when the vector space comprises matrices, such norms are referred to asmatrix norms. Matrix norms differ from vector norms in that they must also interact withmatrix multiplication.

Given afield${\displaystyle K}$ of eitherreal orcomplex numbers (or any complete subset thereof), let${\displaystyle K^{m\times n}}$ be theK-vector space of matrices with${\displaystyle m}$ rows and${\displaystyle n}$ columns and entries in the field${\displaystyle K.}$ A matrix norm is anorm on${\displaystyle K^{m\times n}.}$

Norms are often expressed withdouble vertical bars (like so:${\displaystyle \Vert A\Vert }$). Thus, the matrix norm is afunction${\displaystyle \Vert \cdot \Vert :K^{m\times n}\to {\mathbb{R}}^{0+}}$ that must satisfy the following properties:^[1][2]

For all scalars${\displaystyle \alpha \in K}$ and matrices${\displaystyle A,B\in K^{m\times n},}$

• ${\displaystyle \Vert A\Vert \ge 0}$ (positive-valued)
• ${\displaystyle \Vert A\Vert =0⟺A=0_{m,n}}$ (definite)
• ${\displaystyle \Vert \alpha A\Vert =\left|\alpha \right|\Vert A\Vert }$ (absolutely homogeneous)
• ${\displaystyle \Vert A+B\Vert \le \Vert A\Vert +\Vert B\Vert }$ (sub-additive or satisfying thetriangle inequality)

The only feature distinguishing matrices from rearranged vectors ismultiplication. Matrix norms are particularly useful if they are alsosub-multiplicative:^[1][2][3]

• ${\displaystyle \Vert AB\Vert \le \Vert A\Vert \Vert B\Vert }$^[a]

Every norm on${\displaystyle K^{n\times n}}$ can be rescaled to be sub-multiplicative; in some books, the terminologymatrix norm is reserved for sub-multiplicative norms.^[4]

A matrix norm is called unitarily invariant if for all unitary matrices${\displaystyle U,V}$ and matrix${\displaystyle A}$,${\displaystyle \Vert UAV\Vert =\Vert A\Vert }$.

A symmetric gauge function is an absolutevector norm${\displaystyle \varphi :{\mathbb{C}}^p\to {\mathbb{R}}^{+}}$ such that${\displaystyle \varphi (Px)=\varphi (x)}$ for anypermutation matrix${\displaystyle P}$. That is:

• Non-negativity:${\displaystyle \varphi (x)\ge 0}$, and${\displaystyle \varphi (x)=0}$ if and only if${\displaystyle x=0}$.
• Positive homogeneity:${\displaystyle \varphi (\alpha x)=|\alpha |\varphi (x)}$ for any real number${\displaystyle \alpha }$.
• Triangle inequality:${\displaystyle \varphi (x+y)\le \varphi (x)+\varphi (y)}$.
• Symmetry:${\displaystyle \varphi (Px)=\varphi (x)}$ for any permutation matrix${\displaystyle P}$.

A norm is a unitarily invariant matrix normif and only if it is a symmetric gauge function on the vector of singular values.^[4]

Suppose avector norm${\displaystyle \Vert \cdot {\Vert }_{\alpha }}$ on${\displaystyle K^n}$ and a vector norm${\displaystyle \Vert \cdot {\Vert }_{\beta }}$ on${\displaystyle K^m}$ are given. Any${\displaystyle m\times n}$ matrixA induces a linear operator from${\displaystyle K^n}$ to${\displaystyle K^m}$ with respect to the standard basis, and one defines the correspondinginduced norm oroperator norm orsubordinate norm on the space${\displaystyle K^{m\times n}}$ of all${\displaystyle m\times n}$ matrices as follows:$${\displaystyle \Vert A{\Vert }_{\alpha ,\beta }=sup\{\Vert Ax{\Vert }_{\beta }:x\in K^n\text{ such that }\Vert x{\Vert }_{\alpha }\le 1\}}$$where${\displaystyle sup}$ denotes thesupremum. This norm measures how much the mapping induced by${\displaystyle A}$ can stretch vectors.Depending on the vector norms${\displaystyle \Vert \cdot {\Vert }_{\alpha }}$,${\displaystyle \Vert \cdot {\Vert }_{\beta }}$ used, notation other than${\displaystyle \Vert \cdot {\Vert }_{\alpha ,\beta }}$ can be used for the operator norm.

If thep-norm for vectors (${\displaystyle 1\le p\le \mathrm{\infty }}$) is used for both spaces${\displaystyle K^n}$ and${\displaystyle K^m,}$ then the corresponding operator norm is:^[2]$${\displaystyle \Vert A{\Vert }_p=sup\{\Vert Ax{\Vert }_p:x\in K^n\text{ such that }\Vert x{\Vert }_p\le 1\}.}$$These induced norms are different from the"entry-wise"p-norms and theSchattenp-norms for matrices treated below, which are also usually denoted by${\displaystyle \Vert A{\Vert }_p.}$

Geometrically speaking, one can imagine ap-norm unit ball${\displaystyle V_{p,n}=\{x\in K^n:\Vert x{\Vert }_p\le 1\}}$ in${\displaystyle K^n}$, then apply the linear map${\displaystyle A}$ to the ball. It would end up becoming a distorted convex shape${\displaystyle A{V}_{p,n}\subset K^m}$, and${\displaystyle \Vert A{\Vert }_p}$ measures the longest "radius" of the distorted convex shape. In other words, we must take ap-norm unit ball${\displaystyle V_{p,m}}$ in${\displaystyle K^m}$, then multiply it by at least${\displaystyle \Vert A{\Vert }_p}$, in order for it to be large enough to contain${\displaystyle A{V}_{p,n}}$.

When${\displaystyle p=1,}$ or${\displaystyle p=\mathrm{\infty },}$ we have simple formulas.

$${\displaystyle \Vert A{\Vert }_1=\underset{1\le j\le n}{max}\sum _{i=1}^m\left|a_{ij}\right|,}$$

which is simply the maximum absolute column sum of the matrix.$${\displaystyle \Vert A{\Vert }_{\mathrm{\infty }}=\underset{1\le i\le m}{max}\sum _{j=1}^n\left|a_{ij}\right|,}$$which is simply the maximum absolute row sum of the matrix.

For example, forwe have that$${\displaystyle \Vert A{\Vert }_1=max\{|-3|+2+0,5+6+2,7+4+8\}=max\{5,13,19\}=19,}$$$${\displaystyle \Vert A{\Vert }_{\mathrm{\infty }}=max\{|-3|+5+7,2+6+4,0+2+8\}=max\{15,12,10\}=15.}$$

When${\displaystyle p=2}$ (theEuclidean norm or${\displaystyle {\ell }_2}$-norm for vectors), the induced matrix norm is thespectral norm. The two values donot coincide in infinite dimensions — seeSpectral radius for further discussion. The spectral radius should not be confused with the spectral norm. The spectral norm of a matrix${\displaystyle A}$ is the largestsingular value of${\displaystyle A}$, i.e., the square root of the largesteigenvalue of the matrix${\displaystyle A^{\ast }A,}$ where${\displaystyle A^{\ast }}$ denotes theconjugate transpose of${\displaystyle A}$:^[5]$${\displaystyle \Vert A{\Vert }_2=\sqrt{{\lambda }_{max}\left(A^{\ast }A\right)}={\sigma }_{max}(A).}$$where${\displaystyle {\sigma }_{max}(A)}$ represents the largest singular value of matrix${\displaystyle A.}$

There are further properties:

• $\Vert A{\Vert }_2=sup\{x^{\ast }Ay:x\in K^m,y\in K^n\text{ with }\Vert x{\Vert }_2=\Vert y{\Vert }_2=1\}.$ Proved by theCauchy–Schwarz inequality.
• $\Vert A^{\ast }A{\Vert }_2=\Vert A{A}^{\ast }{\Vert }_2=\Vert A{\Vert }_2^2$. Proven bysingular value decomposition (SVD) on${\displaystyle A}$.
• $\Vert A{\Vert }_2={\sigma }_{\mathrm{m}\mathrm{a}\mathrm{x}}(A)\le \Vert A{\Vert }_{\mathrm{F}}=\sqrt{\sum _i{\sigma }_i(A{)}^2}$, where${\displaystyle \Vert A{\Vert }_{\text{F}}}$ is theFrobenius norm. Equality holds if and only if the matrix${\displaystyle A}$ is a rank-one matrix or a zero matrix.
• Conversely,${\displaystyle \Vert A{\Vert }_{\text{F}}\le min(m,n{)}^{1/2}\Vert A{\Vert }_2}$.

• ${\displaystyle \Vert A{\Vert }_2=\sqrt{\rho (A^{\ast }A)}\le \sqrt{\Vert A^{\ast }A{\Vert }_{\mathrm{\infty }}}\le \sqrt{\Vert A{\Vert }_1\Vert A{\Vert }_{\mathrm{\infty }}}}$.

We can generalize the above definition. Suppose we have vector norms${\displaystyle \Vert \cdot {\Vert }_{\alpha }}$ and${\displaystyle \Vert \cdot {\Vert }_{\beta }}$ for spaces${\displaystyle K^n}$ and${\displaystyle K^m}$ respectively; the corresponding operator norm is$${\displaystyle \Vert A{\Vert }_{\alpha ,\beta }=sup\{\Vert Ax{\Vert }_{\beta }:x\in K^n\text{ such that }\Vert x{\Vert }_{\alpha }\le 1\}}$$In particular, the${\displaystyle \Vert A{\Vert }_p}$ defined previously is the special case of${\displaystyle \Vert A{\Vert }_{p,p}}$.

In the special cases of${\displaystyle \alpha =2}$ and${\displaystyle \beta =\mathrm{\infty }}$, the induced matrix norms can be computed by$${\displaystyle \Vert A{\Vert }_{2,\mathrm{\infty }}=\underset{1\le i\le m}{max}\Vert A_{i:}{\Vert }_2,}$$ where${\displaystyle A_{i:}}$ is the i-th row of matrix${\displaystyle A}$.

In the special cases of${\displaystyle \alpha =1}$ and${\displaystyle \beta =2}$, the induced matrix norms can be computed by$${\displaystyle \Vert A{\Vert }_{1,2}=\underset{1\le j\le n}{max}\Vert A_{:j}{\Vert }_2,}$$ where${\displaystyle A_{:j}}$ is the j-th column of matrix${\displaystyle A}$.

Hence,${\displaystyle \Vert A{\Vert }_{2,\mathrm{\infty }}}$ and${\displaystyle \Vert A{\Vert }_{1,2}}$ are the maximum row and column 2-norm of the matrix, respectively.

Any operator norm isconsistent with the vector norms that induce it, giving$${\displaystyle \Vert Ax{\Vert }_{\beta }\le \Vert A{\Vert }_{\alpha ,\beta }\Vert x{\Vert }_{\alpha }.}$$

Suppose${\displaystyle \Vert \cdot {\Vert }_{\alpha ,\beta }}$;${\displaystyle \Vert \cdot {\Vert }_{\beta ,\gamma }}$; and${\displaystyle \Vert \cdot {\Vert }_{\alpha ,\gamma }}$ are operator norms induced by the respective pairs of vector norms${\displaystyle (\Vert \cdot {\Vert }_{\alpha },\Vert \cdot {\Vert }_{\beta })}$;${\displaystyle (\Vert \cdot {\Vert }_{\beta },\Vert \cdot {\Vert }_{\gamma })}$; and${\displaystyle (\Vert \cdot {\Vert }_{\alpha },\Vert \cdot {\Vert }_{\gamma })}$. Then,

${\displaystyle \Vert AB{\Vert }_{\alpha ,\gamma }\le \Vert A{\Vert }_{\beta ,\gamma }\Vert B{\Vert }_{\alpha ,\beta };}$

this follows from$${\displaystyle \Vert ABx{\Vert }_{\gamma }\le \Vert A{\Vert }_{\beta ,\gamma }\Vert Bx{\Vert }_{\beta }\le \Vert A{\Vert }_{\beta ,\gamma }\Vert B{\Vert }_{\alpha ,\beta }\Vert x{\Vert }_{\alpha }}$$and$${\displaystyle \underset{\Vert x{\Vert }_{\alpha }=1}{sup}\Vert ABx{\Vert }_{\gamma }=\Vert AB{\Vert }_{\alpha ,\gamma }.}$$

Suppose${\displaystyle \Vert \cdot {\Vert }_{\alpha ,\alpha }}$ is an operator norm on the space of square matrices${\displaystyle K^{n\times n}}$induced by vector norms${\displaystyle \Vert \cdot {\Vert }_{\alpha }}$ and${\displaystyle \Vert \cdot {\Vert }_{\alpha }}$.Then, the operator norm is a sub-multiplicative matrix norm:$${\displaystyle \Vert AB{\Vert }_{\alpha ,\alpha }\le \Vert A{\Vert }_{\alpha ,\alpha }\Vert B{\Vert }_{\alpha ,\alpha }.}$$

Moreover, any such norm satisfies the inequality

$${\displaystyle (\Vert A^r{\Vert }_{\alpha ,\alpha }{)}^{1/r}\ge \rho (A)}$$

1

for all positive integersr, whereρ(A) is thespectral radius ofA. Forsymmetric orhermitianA, we have equality in (1) for the 2-norm, since in this case the 2-normis precisely the spectral radius ofA. For an arbitrary matrix, we may not have equality for any norm; a counterexample would bewhich has vanishing spectral radius. In any case, for any matrix norm, we have thespectral radius formula:$${\displaystyle \underset{r\to \mathrm{\infty }}{lim}\Vert A^r{\Vert }^{1/r}=\rho (A).}$$

If the vector norms${\displaystyle \Vert \cdot {\Vert }_{\alpha }}$ and${\displaystyle \Vert \cdot {\Vert }_{\beta }}$ are given in terms ofenergy norms based onsymmetricpositive definite matrices${\displaystyle P}$ and${\displaystyle Q}$ respectively, the resulting operator norm is given as$${\displaystyle \Vert A{\Vert }_{P,Q}=sup\{\Vert Ax{\Vert }_Q:\Vert x{\Vert }_P\le 1\}.}$$

Using the symmetricmatrix square roots of${\displaystyle P}$ and${\displaystyle Q}$ respectively, the operator norm can be expressed as the spectral norm of a modified matrix:

$${\displaystyle \Vert A{\Vert }_{P,Q}=\Vert Q^{1/2}A{P}^{-1/2}{\Vert }_2.}$$

A matrix norm${\displaystyle \Vert \cdot \Vert }$ on${\displaystyle K^{m\times n}}$ is calledconsistent with a vector norm${\displaystyle \Vert \cdot {\Vert }_{\alpha }}$ on${\displaystyle K^n}$ and a vector norm${\displaystyle \Vert \cdot {\Vert }_{\beta }}$ on${\displaystyle K^m}$, if:$${\displaystyle {\Vert Ax\Vert }_{\beta }\le \Vert A\Vert {\Vert x\Vert }_{\alpha }}$$for all${\displaystyle A\in K^{m\times n}}$ and all${\displaystyle x\in K^n}$. In the special case ofm =n and${\displaystyle \alpha =\beta }$,${\displaystyle \Vert \cdot \Vert }$ is also calledcompatible with${\displaystyle \Vert \cdot {\Vert }_{\alpha }}$.

All induced norms are consistent by definition. Also, any sub-multiplicative matrix norm on${\displaystyle K^{n\times n}}$ induces a compatible vector norm on${\displaystyle K^n}$ by defining${\displaystyle \Vert v\Vert :=\Vert \left(v,v,\dots ,v\right)\Vert }$.

These norms treat an${\displaystyle m\times n}$ matrix as a vector of size${\displaystyle m\cdot n}$, and use one of the familiar vector norms. For example, using thep-norm for vectors,p ≥ 1, we get:

${\displaystyle \Vert A{\Vert }_{p,p}=\Vert \mathrm{v}\mathrm{e}\mathrm{c}(A){\Vert }_p={\left(\sum _{i=1}^m\sum _{j=1}^n|a_{ij}{|}^p\right)}^{1/p}}$

This is a different norm from the inducedp-norm (see above) and the Schattenp-norm (see below), but the notation is the same.

The special casep = 2 is the Frobenius norm, andp = ∞ yields the maximum norm.

Let${\displaystyle (a_1,\dots ,a_n)}$ be the dimensionm columns of matrix${\displaystyle A}$. From the original definition, the matrix${\displaystyle A}$ presentsn data points in anm-dimensional space. The${\displaystyle L_{2,1}}$ norm^[6] is the sum of the Euclidean norms of the columns of the matrix:

${\displaystyle \Vert A{\Vert }_{2,1}=\sum _{j=1}^n\Vert a_j{\Vert }_2=\sum _{j=1}^n{\left(\sum _{i=1}^m|a_{ij}{|}^2\right)}^{1/2}}$

The${\displaystyle L_{2,1}}$ norm as anerror function is more robust, since the error for each data point (a column) is not squared. It is used inrobust data analysis andsparse coding.

Forp,q ≥ 1, the${\displaystyle L_{2,1}}$ norm can be generalized to the${\displaystyle L_{p,q}}$ norm as follows:

${\displaystyle \Vert A{\Vert }_{p,q}={\left(\sum _{j=1}^n{\left(\sum _{i=1}^m|a_{ij}{|}^p\right)}^{\frac{q}{p}}\right)}^{\frac{1}{q}}.}$

Whenp =q = 2 for the${\displaystyle L_{p,q}}$ norm, it is called theFrobenius norm or theHilbert–Schmidt norm, though the latter term is used more frequently in the context of operators on (possibly infinite-dimensional)Hilbert space. This norm can be defined in various ways:

${\displaystyle \Vert A{\Vert }_{\text{F}}=\sqrt{\sum _i^m\sum _j^n|a_{ij}{|}^2}=\sqrt{\mathrm{trace}\left(A^{\ast }A\right)}=\sqrt{\sum _{i=1}^{min\{m,n\}}{\sigma }_i^2(A)},}$

where thetrace is the sum of diagonal entries, and${\displaystyle {\sigma }_i(A)}$ are thesingular values of${\displaystyle A}$. The second equality is proven by explicit computation of${\displaystyle \mathrm{t}\mathrm{r}\mathrm{a}\mathrm{c}\mathrm{e}(A^{\ast }A)}$. The third equality is proven bysingular value decomposition of${\displaystyle A}$, and the fact that the trace is invariant under circular shifts.

The Frobenius norm is an extension of the Euclidean norm to${\displaystyle K^{n\times n}}$ and comes from theFrobenius inner product on the space of all matrices.

The Frobenius norm is sub-multiplicative and is very useful fornumerical linear algebra. The sub-multiplicativity of Frobenius norm can be proved using theCauchy–Schwarz inequality. In fact, it is more than sub-multiplicative, as$${\displaystyle \Vert AB{\Vert }_F\le \Vert A{\Vert }_{op}\Vert B{\Vert }_F}$$where the operator norm${\displaystyle \Vert \cdot {\Vert }_{op}\le \Vert \cdot {\Vert }_F}$.

Frobenius norm is often easier to compute than induced norms, and has the useful property of being invariant underrotations (andunitary operations in general). That is,${\displaystyle \Vert A{\Vert }_{\text{F}}=\Vert AU{\Vert }_{\text{F}}=\Vert UA{\Vert }_{\text{F}}}$ for any unitary matrix${\displaystyle U}$. This property follows from the cyclic nature of the trace (${\displaystyle \mathrm{trace}(XYZ)=\mathrm{trace}(YZX)=\mathrm{trace}(ZXY)}$):

${\displaystyle \Vert AU{\Vert }_{\text{F}}^2=\mathrm{trace}\left((AU{)}^{\ast }AU\right)=\mathrm{trace}\left(U^{\ast }{A}^{\ast }AU\right)=\mathrm{trace}\left(U{U}^{\ast }{A}^{\ast }A\right)=\mathrm{trace}\left(A^{\ast }A\right)=\Vert A{\Vert }_{\text{F}}^2,}$

and analogously:

${\displaystyle \Vert UA{\Vert }_{\text{F}}^2=\mathrm{trace}\left((UA{)}^{\ast }UA\right)=\mathrm{trace}\left(A^{\ast }{U}^{\ast }UA\right)=\mathrm{trace}\left(A^{\ast }A\right)=\Vert A{\Vert }_{\text{F}}^2,}$

where we have used the unitary nature of${\displaystyle U}$ (that is,${\displaystyle U^{\ast }U=U{U}^{\ast }=\mathbf{I}}$).

It also satisfies

${\displaystyle \Vert A^{\ast }A{\Vert }_{\text{F}}=\Vert A{A}^{\ast }{\Vert }_{\text{F}}\le \Vert A{\Vert }_{\text{F}}^2}$

and

${\displaystyle \Vert A+B{\Vert }_{\text{F}}^2=\Vert A{\Vert }_{\text{F}}^2+\Vert B{\Vert }_{\text{F}}^2+2\mathrm{Re}\left(⟨A,B{⟩}_{\text{F}}\right),}$

where${\displaystyle ⟨A,B{⟩}_{\text{F}}}$ is theFrobenius inner product, and Re is the real part of a complex number (irrelevant for real matrices)

Themax norm is the elementwise norm in the limit asp =q goes to infinity:

${\displaystyle \Vert A{\Vert }_{max}=\underset{i,j}{max}|a_{ij}|.}$

This norm is notsub-multiplicative; but modifying the right-hand side to${\displaystyle \sqrt{mn}\underset{i,j}{max}|a_{ij}|}$ makes it so.

Note that in some literature (such asCommunication complexity), an alternative definition of max-norm, also called the${\displaystyle {\gamma }_2}$-norm, refers to the factorization norm:

${\displaystyle {\gamma }_2(A)=\underset{U,V:A=U{V}^T}{min}\Vert U{\Vert }_{2,\mathrm{\infty }}\Vert V{\Vert }_{2,\mathrm{\infty }}=\underset{U,V:A=U{V}^T}{min}\underset{i,j}{max}\Vert U_{i,:}{\Vert }_2\Vert V_{j,:}{\Vert }_2}$

The Schattenp-norms arise when applying thep-norm to the vector ofsingular values of a matrix.^[2] If the singular values of the${\displaystyle m\times n}$ matrix${\displaystyle A}$ are denoted byσ_i, then the Schattenp-norm is defined by

${\displaystyle \Vert A{\Vert }_p={\left(\sum _{i=1}^{min\{m,n\}}{\sigma }_i^p(A)\right)}^{1/p}.}$

These norms again share the notation with the induced and entry-wisep-norms, but they are different.

All Schatten norms are sub-multiplicative. They are also unitarily invariant, which means that${\displaystyle \Vert A\Vert =\Vert UAV\Vert }$ for all matrices${\displaystyle A}$ and allunitary matrices${\displaystyle U}$ and${\displaystyle V}$.

The most familiar cases arep = 1, 2, ∞. The casep = 2 yields the Frobenius norm, introduced before. The casep = ∞ yields the spectral norm, which is the operator norm induced by the vector 2-norm (see above). Finally,p = 1 yields thenuclear norm (also known as thetrace norm, or theKy Fan 'n'-norm^[7]), defined as:

${\displaystyle \Vert A{\Vert }_{\ast }=\mathrm{trace}\left(\sqrt{A^{\ast }A}\right)=\sum _{i=1}^{min\{m,n\}}{\sigma }_i(A),}$

where${\displaystyle \sqrt{A^{\ast }A}}$ denotes a positive semidefinite matrix${\displaystyle B}$ such that${\displaystyle BB=A^{\ast }A}$. More precisely, since${\displaystyle A^{\ast }A}$ is apositive semidefinite matrix, itssquare root is well defined. The nuclear norm${\displaystyle \Vert A{\Vert }_{\ast }}$ is aconvex envelope of the rank function${\displaystyle \text{rank}(A)}$, so it is often used inmathematical optimization to search for low-rank matrices.

Combiningvon Neumann's trace inequality withHölder's inequality for Euclidean space yields a version ofHölder's inequality for Schatten norms for${\displaystyle 1/p+1/q=1}$:

${\displaystyle \left|\mathrm{trace}(A^{\ast }B)\right|\le \Vert A{\Vert }_p\Vert B{\Vert }_q,}$

In particular, this implies the Schatten norm inequality

${\displaystyle \Vert A{\Vert }_F^2\le \Vert A{\Vert }_p\Vert A{\Vert }_q.}$

A matrix norm${\displaystyle \Vert \cdot \Vert }$ is calledmonotone if it is monotonic with respect to theLoewner order. Thus, a matrix norm is increasing if

${\displaystyle A\preccurlyeq B\Rightarrow \Vert A\Vert \le \Vert B\Vert .}$

The Frobenius norm and spectral norm are examples of monotone norms.^[8]

Another source of inspiration for matrix norms arises from considering a matrix as theadjacency matrix of aweighted,directed graph.^[9] The so-called "cut norm" measures how close the associated graph is to beingbipartite:$${\displaystyle \Vert A{\Vert }_{◻}=\underset{S\subseteq [n],T\subseteq [m]}{max}\left|\sum _{s\in S,t\in T}{A}_{t,s}\right|}$$ whereA ∈K^m×n.^[9][10][11] Equivalent definitions (up to a constant factor) impose the conditions2|S| >n & 2|T| >m;S =T; orS ∩T = ∅.^[10]

The cut-norm is equivalent to the induced operator norm‖·‖_∞→1, which is itself equivalent to another norm, called theGrothendieck norm.^[11]

To define the Grothendieck norm, first note that a linear operatorK¹ →K¹ is just a scalar, and thus extends to a linear operator on anyK^k →K^k. Moreover, given any choice of basis forKⁿ andK^m, any linear operatorKⁿ →K^m extends to a linear operator(K^k)ⁿ → (K^k)^m, by letting each matrix element on elements ofK^k viascalar multiplication. The Grothendieck norm is the norm of that extended operator; in symbols:^[11]$${\displaystyle \Vert A{\Vert }_{G,k}=\underset{\text{each }{u}_j,v_j\in K^k;\Vert u_j\Vert =\Vert v_j\Vert =1}{sup}\sum _{j\in [n],\ell \in [m]}(u_j\cdot v_j)A_{\ell ,j}}$$

The Grothendieck norm depends on choice of basis (usually taken to be thestandard basis) andk.

For any two matrix norms${\displaystyle \Vert \cdot {\Vert }_{\alpha }}$ and${\displaystyle \Vert \cdot {\Vert }_{\beta }}$, we have that:

${\displaystyle r\Vert A{\Vert }_{\alpha }\le \Vert A{\Vert }_{\beta }\le s\Vert A{\Vert }_{\alpha }}$

for some positive numbersr ands, for all matrices${\displaystyle A\in K^{m\times n}}$. In other words, all norms on${\displaystyle K^{m\times n}}$ areequivalent; they induce the sametopology on${\displaystyle K^{m\times n}}$. This is true because the vector space${\displaystyle K^{m\times n}}$ has the finitedimension${\displaystyle m\times n}$.

Moreover, for every matrix norm${\displaystyle \Vert \cdot \Vert }$ on${\displaystyle {\mathbb{R}}^{n\times n}}$ there exists a unique positive real number${\displaystyle k}$ such that${\displaystyle \ell \Vert \cdot \Vert }$ is a sub-multiplicative matrix norm for every${\displaystyle \ell \ge k}$; to wit,

${\displaystyle k=sup\{\Vert AB\Vert :\Vert A\Vert \le 1,\Vert B\Vert \le 1\}.}$

A sub-multiplicative matrix norm${\displaystyle \Vert \cdot {\Vert }_{\alpha }}$ is said to beminimal, if there exists no other sub-multiplicative matrix norm${\displaystyle \Vert \cdot {\Vert }_{\beta }}$ satisfying.

Let${\displaystyle \Vert A{\Vert }_p}$ once again refer to the norm induced by the vectorp-norm (as above in the Induced norm section).

For matrix${\displaystyle A\in {\mathbb{R}}^{m\times n}}$ ofrank${\displaystyle r}$, the following inequalities hold:^[12][13]

• ${\displaystyle \Vert A{\Vert }_2\le \Vert A{\Vert }_F\le \sqrt{r}\Vert A{\Vert }_2}$
• ${\displaystyle \Vert A{\Vert }_F\le \Vert A{\Vert }_{\ast }\le \sqrt{r}\Vert A{\Vert }_F}$
• ${\displaystyle \Vert A{\Vert }_{max}\le \Vert A{\Vert }_2\le \sqrt{mn}\Vert A{\Vert }_{max}}$
• ${\displaystyle \frac{1}{\sqrt{n}}\Vert A{\Vert }_{\mathrm{\infty }}\le \Vert A{\Vert }_2\le \sqrt{m}\Vert A{\Vert }_{\mathrm{\infty }}}$
• ${\displaystyle \frac{1}{\sqrt{m}}\Vert A{\Vert }_1\le \Vert A{\Vert }_2\le \sqrt{n}\Vert A{\Vert }_1.}$

• Dual norm
• Logarithmic norm

• James W. Demmel, Applied Numerical Linear Algebra, section 1.7, published by SIAM, 1997.
• Carl D. Meyer, Matrix Analysis and Applied Linear Algebra, published by SIAM, 2000.[1]
• John Watrous, Theory of Quantum Information,2.3 Norms of operators, lecture notes, University of Waterloo, 2011.
• Kendall Atkinson, An Introduction to Numerical Analysis, published by John Wiley & Sons, Inc 1989

• Norms (mathematics)
• Linear algebra

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