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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).

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DefinitionEdit

In what follows,   will denote a field of either real or complex numbers.

Let   denote the vector space of all matrices of size   (with   rows and   columns) with entries in the field  .

A matrix norm is a norm on the vector space  . Thus, the matrix norm is a function   that must satisfy the following properties:

For all scalars   and for all matrices  ,

  •   (being absolutely homogeneous)
  •   (being sub-additive or satisfying the triangle inequality)
  •   (being positive-valued)
  •   iff   (being definite)

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:

  •   for all matrices   and   in  

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   matrices, together with such a sub-multiplicative norm, is an example of a Banach algebra.

The definition of sub-multiplicativity is sometimes extended to non-square matrices, for instance in the case of the induced p-norm, where for   and   holds that  . Here   and   are the norms induced from   and  , respectively, and p,q ≥ 1.

There are three types of matrix norms which will be discussed below:

  • Matrix norms induced by vector norms,
  • Entrywise matrix norms, and
  • Schatten norms.

Matrix norms induced by vector normsEdit

Suppose 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 on the space   of all   matrices as follows:

 

In particular, if the p-norm for vectors (1 ≤ p ≤ ∞) is used for both spaces   and  , then the corresponding induced operator norm is:

 

These induced norms are different from the "entrywise" p-norms and the Schatten p-norms for matrices treated below, which are also usually denoted by  

Note: We have described above the induced operator norm when the same vector norm was used in the "departure space"   and the "arrival space"   of the operator  . This is not a necessary restriction. More generally, given a norm   on  , and a norm   on  , one can define a matrix norm on   induced by these norms:
 
The matrix norm   is sometimes called a subordinate norm. Subordinate norms are consistent with the norms that induce them, giving
 

Any induced operator norm is a sub-multiplicative matrix norm:   this follows from

 

and

 

Moreover, any induced norm satisfies the inequality

 

 

 

 

 

(1)

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 being given by

 

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

 

Special casesEdit

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;

 

where   represents the largest singular value of matrix  . There is an important inequality for the case  :

 

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.

For example, for

 

we have

 
 

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  :[1]

 

"Entrywise" matrix normsEdit

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 normsEdit

Let   be the columns of matrix  . The   norm[2] 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.

The   norm can be generalized to the   norm, p, q ≥ 1, defined by

 

Frobenius normEdit

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. This norm is often easier to compute than induced norms and has the useful property of being invariant under rotations and, more generally, under unitary operations , 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 normEdit

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 normsEdit

The Schatten p-norms arise when applying the p-norm to the vector of singular values of a matrix. 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 entrywise 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[3]), defined as

 

(Here   denotes a positive semidefinite matrix   such that  . More precisely, since   is a positive semidefinite matrix, its square root is well-defined.)

Consistent normsEdit

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

 

for all  . All induced norms are consistent by definition.

Compatible normsEdit

A matrix norm   on   is called compatible with a vector norm   on   if:

 

for all  . Induced norms are compatible with the inducing vector norm by definition.

Equivalence of normsEdit

For any two matrix norms   and  , we have

 

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 equivalenceEdit

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:[4][5]

  •  
  •  
  •  
  •  
  •  

Another useful inequality between matrix norms is

 

which is a special case of Hölder's inequality.

NotesEdit

  1. ^ Carl D. Meyer, Matrix Analysis and Applied Linear Algebra, §5.2, p.281, Society for Industrial & Applied Mathematics, June 2000.
  2. ^ 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.
  3. ^ 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. doi:10.1073/pnas.37.11.760. PMC 1063464.
  4. ^ Golub, Gene; Charles F. Van Loan (1996). Matrix Computations – Third Edition. Baltimore: The Johns Hopkins University Press, 56–57. ISBN 0-8018-5413-X.
  5. ^ Roger Horn and Charles Johnson. Matrix Analysis, Chapter 5, Cambridge University Press, 1985. ISBN 0-521-38632-2.

ReferencesEdit

  • 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