Matrix norm

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In mathematics, a matrix norm is a vector norm in a vector space whose elements (vectors) are matrices (of given dimensions).

Preliminaries[edit]

Given a field of either real or complex numbers, let be the K-vector space of matrices with rows and columns and entries in the field . A matrix norm is a norm on

This article will always write such norms with double vertical bars (like so: ). Thus, the matrix norm is a function that must satisfy the following properties:[1][2]

For all scalars and matrices ,
  • (positive-valued)
  • (definite)
  • (absolutely homogeneous)
  • (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]

  • [Note 1]

Every norm on Kn×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[edit]

Suppose a vector norm on and a vector norm on is given. Any matrix A induces a linear operator from to with respect to the standard basis, and one defines the corresponding induced norm or operator norm or subordinate norm on the space of all matrices as follows:

Depending on the vector norms , used, notation other than can be used for the operator norm.

Norms induced by p-norms[edit]

If the p-norm for vectors (1 ≤ p ≤ ∞) is used for both spaces and , 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

In the special cases of the induced matrix norms can be computed or estimated by

which is simply the maximum absolute column sum of the matrix;

which is simply the maximum absolute row sum of the matrix.

In the special case of (the Euclidean norm or -norm for vectors), the induced matrix norm is the spectral norm. The spectral norm of a matrix is the largest singular value of (i.e., the square root of the largest eigenvalue of the matrix , where denotes the conjugate transpose of ):[5]

where represents the largest singular value of matrix . Also,

since and similarly by singular value decomposition (SVD). There is another important inequality:

where is the Frobenius norm. Equality holds if and only if the matrix is a rank-one matrix or a zero matrix. This inequality can be derived from the fact that the trace of a matrix is equal to the sum of its eigenvalues.

When we have an equivalent definition for as . It can be shown to be equivalent to the above definitions using the Cauchy–Schwarz inequality.

For example, for

we have that

Properties[edit]

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

Suppose is an operator norm induced by vector norms and . Suppose is an operator norm induced by vector norms and . Suppose is an operator norm induced by vector norms and . Then,

this follows from

and

Square matrices[edit]

Suppose is an operator norm on the space of square matrices induced by vector norms and . Then, the operator norm is a sub-multiplicative matrix norm:

Moreover, any such norm satisfies the inequality

(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[edit]

A matrix norm on is called consistent with a vector norm on and a vector norm on , if:

for all and all . In the special case of m = n and , is also called compatible with .

All induced norms are consistent by definition. Also, any sub-multiplicative matrix norm on induces a compatible vector norm on by defining .

"Entry-wise" matrix norms[edit]

These norms treat an matrix as a vector of size , and use one of the familiar vector norms. For example, using the p-norm for vectors, p ≥ 1, we get:

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 and Lp,q norms[edit]

Let be the columns of matrix . The norm[6] is the sum of the Euclidean norms of the columns of the matrix:

The 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 norm can be generalized to the norm as follows:

Frobenius norm[edit]

When p = q = 2 for the 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:

where are the singular values of . Recall that the trace function returns the sum of diagonal entries of a square matrix.

The Frobenius norm is an extension of the Euclidean norm to 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, for any unitary matrix . This property follows from the cyclic nature of the trace ():

and analogously:

where we have used the unitary nature of (that is, ).

It also satisfies

and

where is the Frobenius inner product.

Max norm[edit]

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

This norm is not sub-multiplicative.

Note that in some literature (such as Communication complexity), an alternative definition of max-norm, also called the -norm, refers to the factorization norm:

Schatten norms[edit]

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 matrix are denoted by σi, then the Schatten p-norm is defined by

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 for all matrices and all unitary matrices and .

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

where denotes a positive semidefinite matrix such that . More precisely, since is a positive semidefinite matrix, its square root is well-defined. The nuclear norm is a convex envelope of the rank function , so it is often used in mathematical optimization to search for low rank matrices.

Monotone norms[edit]

A matrix norm is called monotone if it is monotonic with respect to the Loewner order. Thus, a matrix norm is increasing if

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

Cut norms[edit]

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 AKm×n.[9][10][11] Equivalent definitions (up to a constant factor) impose the conditions 2|S|>n & 2|T|>m, S=T, or ST=∅.[10]

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

To define the Grothendieck norm, first note that a linear operator K1K1 is just a scalar, and thus extends to a linear operator on any KkKk. Moreover, given any choice of basis for Kn and Km, any linear operator KnKm extends to a linear operator (Kk)n→(Kk)m, by letting each matrix element on elements of Kk 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[edit]

For any two matrix norms and , we have that:

for some positive numbers r and s, for all matrices . In other words, all norms on are equivalent; they induce the same topology on . This is true because the vector space has the finite dimension .

Moreover, for every vector norm on , there exists a unique positive real number such that is a sub-multiplicative matrix norm for every .

A sub-multiplicative matrix norm is said to be minimal, if there exists no other sub-multiplicative matrix norm satisfying .

Examples of norm equivalence[edit]

Let once again refer to the norm induced by the vector p-norm (as above in the Induced Norm section).

For matrix of rank , the following inequalities hold:[12][13]

Another useful inequality between matrix norms is

which is a special case of Hölder's inequality.

See also[edit]

Notes[edit]

  1. ^ The condition only applies when the product is defined, such as the case of square matrices (m = n).

References[edit]

  1. ^ a b Weisstein, Eric W. "Matrix Norm". mathworld.wolfram.com. Retrieved 2020-08-24.
  2. ^ a b c d "Matrix norms". fourier.eng.hmc.edu. Retrieved 2020-08-24.
  3. ^ Malek-Shahmirzadi, Massoud (1983). "A characterization of certain classes of matrix norms". Linear and Multilinear Algebra. 13 (2): 97–99. doi:10.1080/03081088308817508. ISSN 0308-1087.
  4. ^ Horn, Roger A. (2012). Matrix analysis. Johnson, Charles R. (2nd ed.). Cambridge: Cambridge University Press. pp. 340–341. ISBN 978-1-139-77600-4. OCLC 817236655.
  5. ^ Carl D. Meyer, Matrix Analysis and Applied Linear Algebra, §5.2, p.281, Society for Industrial & Applied Mathematics, June 2000.
  6. ^ Ding, Chris; Zhou, Ding; He, Xiaofeng; Zha, Hongyuan (June 2006). "R1-PCA: Rotational Invariant L1-norm Principal Component Analysis for Robust Subspace Factorization". Proceedings of the 23rd International Conference on Machine Learning. ICML '06. Pittsburgh, Pennsylvania, USA: ACM. pp. 281–288. doi:10.1145/1143844.1143880. ISBN 1-59593-383-2.
  7. ^ Fan, Ky. (1951). "Maximum properties and inequalities for the eigenvalues of completely continuous operators". Proceedings of the National Academy of Sciences of the United States of America. 37 (11): 760–766. Bibcode:1951PNAS...37..760F. doi:10.1073/pnas.37.11.760. PMC 1063464. PMID 16578416.
  8. ^ Ciarlet, Philippe G. (1989). Introduction to numerical linear algebra and optimisation. Cambridge, England: Cambridge University Press. p. 57. ISBN 0521327881.
  9. ^ a b Frieze, Alan; Kannan, Ravi (1999-02-01). "Quick Approximation to Matrices and Applications". Combinatorica. 19 (2): 175–220. doi:10.1007/s004930050052. ISSN 1439-6912. S2CID 15231198.
  10. ^ a b Lovász László (2012). "The cut distance". Large Networks and Graph Limits. AMS Colloquium Publications. 60. Providence, RI: American Mathematical Society. pp. 127–131. ISBN 978-0-8218-9085-1. Note that Lovász rescales A to lie in [0, 1].
  11. ^ a b c Alon, Noga; Naor, Assaf (2004-06-13). "Approximating the cut-norm via Grothendieck's inequality". Proceedings of the Thirty-sixth Annual ACM Symposium on Theory of Computing. STOC '04. Chicago, IL, USA: Association for Computing Machinery: 72–80. doi:10.1145/1007352.1007371. ISBN 978-1-58113-852-8. S2CID 1667427.
  12. ^ Golub, Gene; Charles F. Van Loan (1996). Matrix Computations – Third Edition. Baltimore: The Johns Hopkins University Press, 56–57. ISBN 0-8018-5413-X.
  13. ^ Roger Horn and Charles Johnson. Matrix Analysis, Chapter 5, Cambridge University Press, 1985. ISBN 0-521-38632-2.

Bibliography[edit]

  • 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