- Matrix norm
In what follows, K will denote the field of real or complex numbers. Let denote the vector space containing all matrices with m rows and n columns with entries in K. Throughout the article A * denotes the conjugate transpose of matrix A.
A matrix norm is a vector norm on . That is, if denotes the norm of the matrix A, then,
- if and iff A = 0
- for all α in K and all matrices A in
- for all matrices A and B 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 A and B 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 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:
These are different from the entrywise p-norms and the Schatten p-norms for matrices treated below, which are also usually denoted by
If m = n and one uses the same norm on the domain and the range, then the induced operator norm is a sub-multiplicative matrix norm.
The operator norm corresponding to the p-norm for vectors is:
In the case of p = 1 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(5,13,19) = 19. and ||A||∞ = max(15,12,10) = 15. Consider another example
where we add all the entries in each column and determine the greatest value, which results in ||A||1 = max (6,13,11,8) = 13.
We can do the same for the rows and get ||A||∞ = max(9,11,9,9) = 11. Thus 11 is our max.
In the special case of p = 2 (the Euclidean norm) and m = n (square matrices), 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 Kn, and on Km as:
Subordinate norms are consistent with the norms that induce them, giving
Any induced norm satisfies the inequality
where ρ(A) is the spectral radius of A. For a symmetric or hermitian matrix A, we have equality for the 2-norm, since in this case the 2-norm is the spectral radius of A. For an arbitrary matrix, we may not have equality for any . Take
the spectral radius of A is 0, but A is not the zero matrix, and so none of the induced norms are equal to the spectral radius of A.
Furthermore, for square matrices we have the spectral radius formula:
These vector norms treat an matrix as a vector of size mn, 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.
For p = 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 very similar to the Euclidean norm on Kn and comes from an 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.
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
(Here denotes a matrix B such that BB = A * A. More precisely, since A * A is a positive semidefinite matrix, its square root is well-defined.)
A matrix norm on is called consistent with a vector norm on Kn and a vector norm on Km if:
for all . All induced norms are consistent by definition.
A matrix norm on is called compatible with a vector norm on Kn if:
for all . Induced norms are compatible by definition.
Equivalence of norms
For any two vector norms ||·||α and ||·||β, we have
for some positive numbers r and s, for all matrices A in . In other words, they are equivalent norms; they induce the same topology on . This is a special case of the equivalence of norms in finite-dimensional Normed vector spaces.
Moreover, for every vector norm on , there exists a unique positive real number k such that is a sub-multiplicative matrix norm for every .
A matrix norm ||·||α is said to be minimal if there exists no other matrix norm ||·||β satisfying ||·||β ≤ ||·||α.
Examples of norm equivalence
Here, ||·||p refers to the matrix norm induced by the vector p-norm.
Another useful inequality between matrix norms is
- 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, 2.4 Norms of operators, lecture notes, University of Waterloo, 2008.
- Kendall Atkinson, An Introduction to Numerical Analysis, published by John Wiley & Sons, Inc 1989
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