In what follows, will denote the field of real or complex numbers. Let denote the vector space containing all matrices with rows and columns with entries in . Throughout the article denotes the conjugate transpose of matrix .
A matrix norm is a vector norm on . That is, if denotes the norm of the matrix , then,
- for all in and all matrices in
- for all matrices and in
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 n-by-n matrices, together with such a sub-multiplicative norm, is an example of a Banach algebra.
If vector norms on Km and Kn are given (K is the 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:
The operator norm corresponding to the p-norm for vectors is:
These are different from the entrywise p-norms and the Schatten p-norms for matrices treated below, which are also usually denoted by
In the case of and , the norms can be computed as:
- which is simply the maximum absolute column sum of the matrix.
- which is simply the maximum absolute row sum of the matrix
For example, if the matrix A is defined by
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.
In the special case of p = 2 (the Euclidean norm), 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:
where A* denotes the conjugate transpose of A.
More generally, one can define the subordinate matrix norm on induced by on , and on as:
Subordinate norms are consistent with the norms that induce them, giving
For , any induced operator norm is a sub-multiplicative matrix norm since and .
Any induced norm satisfies the inequality
where ρ(A) is the spectral radius of A. For a symmetric or hermitian matrix , we have equality for the 2-norm, since in this case the 2-norm is the spectral radius of . For an arbitrary matrix, we may not have equality for any norm. Take
the spectral radius of is 0, but is not the zero matrix, and so none of the induced norms are equal to the spectral radius of .
Furthermore, for square matrices we have the spectral radius formula:
Useful estimates of Matrix Norms
It is still an open problem in mathematics to find a formula that gives the norm of the matrix in terms of its entries, however it is not hard to find an expression that is an upper estimate of the matrix norm in terms of its entries. Let be a matrix with complex matrix, and let .
We will denote by
Then the component of Ax is
Then we get
This shows that
These vector 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, 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.
Let be the columns of matrix . The norm is the sum of the Euclidean norm of the columns of the matrix:
In this norm, the two indices and of are treated differently; all matrix norms introduced prior to the norm treat the two indicees symmetrically. The norm is used in robust data analysis and sparse coding for feature selection.
The norm can be generalized to the norm, defined by
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:
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, that is, for any rotation matrix . This property follows from the trace definition restricted to real matrices,
where we have used the orthogonal nature of , that is, , and the cyclic nature of the trace, . More generally the norm is invariant under a unitary transformation for complex matrices.
The max norm is the elementwise norm with p = ∞:
This norm is not sub-multiplicative.
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
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 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.
A matrix norm on is called compatible with a vector norm on if:
for all . Induced norms are compatible by definition.
Equivalence of norms
For any two vector norms and , we have
for some positive numbers r and s, for all matrices A in . 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
Here, refers to the matrix norm induced by the vector p-norm.
Another useful inequality between matrix norms is
which is a special case of Hölder's inequality.
- Carl D. Meyer, Matrix Analysis and Applied Linear Algebra,section 5.2,p281, Society for Industrial & Applied Mathematics,June 2000.
- 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.
- Golub, Gene; Charles F. Van Loan (1996). Matrix Computations - Third Edition. Baltimore: The Johns Hopkins University Press, 56-57. ISBN 0-8018-5413-X.
- Roger Horn and Charles Johnson. Matrix Analysis, Chapter 5, Cambridge University Press, 1985. ISBN 0-521-38632-2.
- 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. 
- 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