Matrix norm

In the field of mathematics, norms are defined for elements within a vector space. Specifically, when the vector space comprises matrices, such norms are referred to as matrix norms. Matrix norms differ from vector norms in that they must also interact with matrix multiplication.

Preliminaries

Given a field ${\displaystyle K}$ of either real or complex numbers, let ${\displaystyle K^{m\times n}}$ be the K-vector space of matrices with ${\displaystyle m}$ rows and ${\displaystyle n}$ columns and entries in the field ${\displaystyle K}$. A matrix norm is a norm on ${\displaystyle K^{m\times n}}$.

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

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

• ${\displaystyle \|A\|\geq 0}$ (positive-valued)
• ${\displaystyle \|A\|=0\iff A=0_{m,n}}$ (definite)
• ${\displaystyle \left\|\alpha A\right\|=\left|\alpha \right|\left\|A\right\|}$ (absolutely homogeneous)
• ${\displaystyle \|A+B\|\leq \|A\|+\|B\|}$ (sub-additive or satisfying the triangle inequality)

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

• ${\displaystyle \left\|AB\right\|\leq \left\|A\right\|\left\|B\right\|}$^[Note 1]

Every norm on K^n×n can be rescaled to be sub-multiplicative; in some books, the terminology matrix norm is reserved for sub-multiplicative norms.^[4]

Matrix norms induced by vector norms

Suppose a vector norm ${\displaystyle \|\cdot \|_{\alpha }}$ on ${\displaystyle K^{n}}$ and a vector norm ${\displaystyle \|\cdot \|_{\beta }}$ on ${\displaystyle K^{m}}$ are given. Any ${\displaystyle m\times n}$ matrix A induces a linear operator from ${\displaystyle K^{n}}$ to ${\displaystyle K^{m}}$ with respect to the standard basis, and one defines the corresponding induced norm or operator norm or subordinate norm on the space ${\displaystyle K^{m\times n}}$ of all ${\displaystyle m\times n}$ matrices as follows:

where ${\displaystyle \sup }$ denotes the supremum. This norm measures how much the mapping induced by ${\displaystyle A}$ can stretch vectors. Depending on the vector norms ${\displaystyle \|\cdot \|_{\alpha }}$, ${\displaystyle \|\cdot \|_{\beta }}$ used, notation other than ${\displaystyle \|\cdot \|_{\alpha ,\beta }}$ can be used for the operator norm.

Matrix norms induced by vector p-norms

If the p-norm for vectors (${\displaystyle 1\leq p\leq \infty }$) is used for both spaces ${\displaystyle K^{n}}$ and ${\displaystyle K^{m},}$ then the corresponding operator norm is:^[2]

These induced norms are different from the "entry-wise" p-norms and the Schatten p-norms for matrices treated below, which are also usually denoted by ${\displaystyle \|A\|_{p}.}$

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

p = 1, ∞

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

which is simply the maximum absolute column sum of the matrix.which is simply the maximum absolute row sum of the matrix. For example, for we have that

Spectral norm (p = 2)

When ${\displaystyle p=2}$ (the Euclidean norm or ${\displaystyle \ell _{2}}$-norm for vectors), the induced matrix norm is the spectral norm. (The two values do not coincide in infinite dimensions — see Spectral radius for further discussion.) The spectral norm of a matrix ${\displaystyle A}$ is the largest singular value of ${\displaystyle A}$ (i.e., the square root of the largest eigenvalue of the matrix ${\displaystyle A^{*}A,}$ where ${\displaystyle A^{*}}$ denotes the conjugate transpose of ${\displaystyle A}$):^[5]

where ${\displaystyle \sigma _{\max }(A)}$ represents the largest singular value of matrix ${\displaystyle A.}$

There are further properties:

• ${\textstyle \|A\|_{2}=\sup\{x^{*}Ay:x\in K^{m},y\in K^{n}{\text{ with }}\|x\|_{2}=\|y\|_{2}=1\}.}$ Proved by the Cauchy–Schwarz inequality.
• ${\textstyle \|A^{*}A\|_{2}=\|AA^{*}\|_{2}=\|A\|_{2}^{2}}$. Proven by singular value decomposition (SVD) on ${\displaystyle A}$.
• ${\textstyle \|A\|_{2}=\sigma _{\mathrm {max} }(A)\leq \|A\|_{\rm {F}}={\sqrt {\sum _{i}\sigma _{i}(A)^{2}}}}$, where ${\displaystyle \|A\|_{\textrm {F}}}$ is the Frobenius norm. Equality holds if and only if the matrix ${\displaystyle A}$ is a rank-one matrix or a zero matrix.

• ${\displaystyle \|A\|_{2}={\sqrt {\rho (A^{*}A)}}\leq {\sqrt {\|A^{*}A\|_{\infty }}}\leq {\sqrt {\|A\|_{1}\|A\|_{\infty }}}}$.

Matrix norms induced by vector α- and β- norms

We can generalize the above definition. Suppose we have vector norms ${\displaystyle \|\cdot \|_{\alpha }}$ and ${\displaystyle \|\cdot \|_{\beta }}$ for spaces ${\displaystyle K^{n}}$ and ${\displaystyle K^{m}}$ respectively; the corresponding operator norm is

In particular, the ${\displaystyle \|A\|_{p}}$ defined previously is the special case of ${\displaystyle \|A\|_{p,p}}$.

In the special cases of ${\displaystyle \alpha =2}$ and ${\displaystyle \beta =\infty }$, the induced matrix norms can be computed by

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

where ${\displaystyle A_{:j}}$ is the j-th column of matrix ${\displaystyle A}$.

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

Properties

Any operator norm is consistent with the vector norms that induce it, giving

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

${\displaystyle \|AB\|_{\alpha ,\gamma }\leq \|A\|_{\beta ,\gamma }\|B\|_{\alpha ,\beta };}$

this follows from

and

Square matrices

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

Moreover, any such norm satisfies the inequality

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

(1)

for all positive integers r, where ρ(A) is the spectral radius of A. For symmetric or hermitian A, we have equality in (1) for the 2-norm, since in this case the 2-norm is precisely the spectral radius of A. For an arbitrary matrix, we may not have equality for any norm; a counterexample would be

which has vanishing spectral radius. In any case, for any matrix norm, we have the spectral radius formula:

Consistent and compatible norms

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

for all ${\displaystyle A\in K^{m\times n}}$ and all ${\displaystyle x\in K^{n}}$. In the special case of m = n and ${\displaystyle \alpha =\beta }$, ${\displaystyle \|\cdot \|}$ is also called compatible with ${\displaystyle \|\cdot \|_{\alpha }}$.

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 ${\displaystyle \left\|v\right\|:=\left\|\left(v,v,\dots ,v\right)\right\|}$.

"Entry-wise" matrix norms

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 the p-norm for vectors, p ≥ 1, we get:

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

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.

L_2,1 and L_p,q norms

Let ${\displaystyle (a_{1},\ldots ,a_{n})}$ be the columns of matrix ${\displaystyle A}$. From the original definition, the matrix ${\displaystyle A}$ presents n data points in m-dimensional space. The ${\displaystyle L_{2,1}}$ norm^[6] is the sum of the Euclidean norms of the columns of the matrix:

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

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

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

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

Frobenius norm

When p = q = 2 for the ${\displaystyle L_{p,q}}$ norm, it is called the Frobenius norm or the Hilbert–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:

${\displaystyle \|A\|_{\text{F}}={\sqrt {\sum _{i}^{m}\sum _{j}^{n}|a_{ij}|^{2}}}={\sqrt {\operatorname {trace} \left(A^{*}A\right)}}={\sqrt {\sum _{i=1}^{\min\{m,n\}}\sigma _{i}^{2}(A)}},}$

where the trace is the sum of diagonal entries, and ${\displaystyle \sigma _{i}(A)}$ are the singular values of ${\displaystyle A}$. The second equality is proven by explicit computation of ${\displaystyle \mathrm {trace} (A^{*}A)}$. The third equality is proven by singular 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 the Frobenius inner product on the space of all matrices.

The Frobenius norm is sub-multiplicative and is very useful for numerical linear algebra. The sub-multiplicativity of Frobenius norm can be proved using Cauchy–Schwarz inequality.

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

${\displaystyle \|AU\|_{\text{F}}^{2}=\operatorname {trace} \left((AU)^{*}AU\right)=\operatorname {trace} \left(U^{*}A^{*}AU\right)=\operatorname {trace} \left(UU^{*}A^{*}A\right)=\operatorname {trace} \left(A^{*}A\right)=\|A\|_{\text{F}}^{2},}$

and analogously:

${\displaystyle \|UA\|_{\text{F}}^{2}=\operatorname {trace} \left((UA)^{*}UA\right)=\operatorname {trace} \left(A^{*}U^{*}UA\right)=\operatorname {trace} \left(A^{*}A\right)=\|A\|_{\text{F}}^{2},}$

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

It also satisfies

${\displaystyle \|A^{*}A\|_{\text{F}}=\|AA^{*}\|_{\text{F}}\leq \|A\|_{\text{F}}^{2}}$

and

${\displaystyle \|A+B\|_{\text{F}}^{2}=\|A\|_{\text{F}}^{2}+\|B\|_{\text{F}}^{2}+2Re\left(\langle A,B\rangle _{\text{F}}\right),}$

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

Max norm

The max norm is the elementwise norm in the limit as p = q goes to infinity:

${\displaystyle \|A\|_{\max }=\max _{ij}|a_{ij}|.}$

This norm is not sub-multiplicative.

Note that in some literature (such as Communication complexity), an alternative definition of max-norm, also called the ${\displaystyle \gamma _{2}}$-norm, refers to the factorization norm:

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

Schatten norms

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

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

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

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

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 operator 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^[7]), defined as:

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

where ${\displaystyle {\sqrt {A^{*}A}}}$ denotes a positive semidefinite matrix ${\displaystyle B}$ such that ${\displaystyle BB=A^{*}A}$. More precisely, since ${\displaystyle A^{*}A}$ is a positive semidefinite matrix, its square root is well defined. The nuclear norm ${\displaystyle \|A\|_{*}}$ is a convex envelope of the rank function ${\displaystyle {\text{rank}}(A)}$, so it is often used in mathematical optimization to search for low-rank matrices.

Combining von Neumann's trace inequality with Hölder's inequality for Euclidean space yields a version of Hölder's inequality for Schatten norms for ${\displaystyle 1/p+1/q=1}$:

${\displaystyle \left|\operatorname {trace} (A'B)\right|\leq \|A\|_{p}\|B\|_{q},}$

In particular, this implies the Schatten norm inequality

${\displaystyle \|A\|_{F}^{2}\leq \|A\|_{p}\|A\|_{q}.}$

Monotone norms

A matrix norm ${\displaystyle \|\cdot \|}$ is called monotone if it is monotonic with respect to the Loewner order. Thus, a matrix norm is increasing if

${\displaystyle A\preccurlyeq B\Rightarrow \|A\|\leq \|B\|.}$

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

Cut norms

Another source of inspiration for matrix norms arises from considering a matrix as the adjacency matrix of a weighted, directed graph.^[9] The so-called "cut norm" measures how close the associated graph is to being bipartite:

where A ∈ K^m×n.^[9][10][11] Equivalent definitions (up to a constant factor) impose the conditions 2|S| > n & 2|T| > m; S = T; or S ∩ T = ∅.^[10]

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

To define the Grothendieck norm, first note that a linear operator K¹ → K¹ is just a scalar, and thus extends to a linear operator on any K^k → K^k. Moreover, given any choice of basis for Kⁿ and K^m, any linear operator Kⁿ → K^m extends to a linear operator (K^k)ⁿ → (K^k)^m, by letting each matrix element on elements of K^k via scalar multiplication. The Grothendieck norm is the norm of that extended operator; in symbols:^[11]

The Grothendieck norm depends on choice of basis (usually taken to be the standard basis) and k.

Equivalence of norms

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

${\displaystyle r\|A\|_{\alpha }\leq \|A\|_{\beta }\leq s\|A\|_{\alpha }}$

for some positive numbers r and s, for all matrices ${\displaystyle A\in K^{m\times n}}$. In other words, all norms on ${\displaystyle K^{m\times n}}$ are equivalent; they induce the same topology on ${\displaystyle K^{m\times n}}$. This is true because the vector space ${\displaystyle K^{m\times n}}$ has the finite dimension ${\displaystyle m\times n}$.

Moreover, for every vector norm ${\displaystyle \|\cdot \|}$ on ${\displaystyle \mathbb {R} ^{n\times n}}$, there exists a unique positive real number ${\displaystyle k}$ such that ${\displaystyle \ell \|\cdot \|}$ is a sub-multiplicative matrix norm for every ${\displaystyle \ell \geq k}$.

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

Examples of norm equivalence

Let ${\displaystyle \|A\|_{p}}$ once again refer to the norm induced by the vector p-norm (as above in the Induced norm section).

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

• ${\displaystyle \|A\|_{2}\leq \|A\|_{F}\leq {\sqrt {r}}\|A\|_{2}}$
• ${\displaystyle \|A\|_{F}\leq \|A\|_{*}\leq {\sqrt {r}}\|A\|_{F}}$
• ${\displaystyle \|A\|_{\max }\leq \|A\|_{2}\leq {\sqrt {mn}}\|A\|_{\max }}$
• ${\displaystyle {\frac {1}{\sqrt {n}}}\|A\|_{\infty }\leq \|A\|_{2}\leq {\sqrt {m}}\|A\|_{\infty }}$
• ${\displaystyle {\frac {1}{\sqrt {m}}}\|A\|_{1}\leq \|A\|_{2}\leq {\sqrt {n}}\|A\|_{1}.}$

See also

• Dual norm
• Logarithmic norm

Notes

References

Bibliography

• 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