In mathematics, the term Matrix Norm can have two meanings: Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ...

• A vector norm on matrices, i.e, a norm on the vector space of all real or complex m-by-n matrices.

• A sub-multiplicative vector norm is any vector norm on square matrices compatible with matrix multiplication in the sense that

The set of all n-by-n matrices, together with such a sub-multiplicative norm, is a Banach algebra.

In the rest of the article, we will follow the tradition in matrix theory. We use term "vector norm" for the first definition and "matrix norm" for the second definition. In mathematics, with 2- or 3-dimensional vectors with real-valued entries, the idea of the length of a vector is intuitive and can easily be extended to any real vector space Rn. ... In mathematics, a vector space (or linear space) is a collection of objects (called vectors) that, informally speaking, may be scaled and added. ... In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ... In mathematics, a complex number is a number of the form where a and b are real numbers, and i is the imaginary unit, with the property i 2 = −1. ... In mathematics, a matrix (plural matrices) is a rectangular table of numbers or, more generally, a table consisting of abstract quantities that can be added and multiplied. ... This article gives an overview of the various ways to perform matrix multiplication. ... This article is about sets in mathematics. ... In functional analysis, a Banach algebra, named after Stefan Banach, is an associative algebra A over the real or complex numbers which at the same time is also a Banach space. ... Matrix theory is a branch of mathematics which focuses on the study of matrices. ...

Operator norm or induced norm

If norms on K^m 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: In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ... In mathematics, a complex number is a number of the form where a and b are real numbers, and i is the imaginary unit, with the property i 2 = −1. ... In mathematics, the operator norm is a means to measure the size of certain linear operators. ...

If m = n and one uses the same norm on domain and range, then these operator norms are all (submultiplicative) matrix norms.

p-norms and uniform norms

For example, the operator norm corresponding to the p-norm for vectors is: In mathematics, with 2- or 3-dimensional vectors with real-valued entries, the idea of the length of a vector is intuitive and can be easily extended to any real vector space Rn. ...

In the case of p = 1 and , the norms can be computed as:

These are different from the Schatten p-norms for matrices, which are also usually denoted by

Spectral norm

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 or the square root of the largest eigenvalue of the positive-semidefinite matrix A^*A: In mathematics and astronomy, Euclidean space is a generalization of the 2- and 3-dimensional spaces studied by Euclid. ... In mathematics, in particular functional analysis, singular values, or s-numbers of an bounded operator T acting on a Hilbert space are defined as the eigenvalues of (T*T)1/2. ... In mathematics, a number is called an eigenvalue of a matrix if there exists a nonzero vector such that the matrix times the vector is equal to the same vector multiplied by the eigenvalue. ... In linear algebra, the positive-definite matrices are (in several ways) analogous to the positive real numbers. ...

where A^* denotes the conjugate transpose of A. In mathematics, the conjugate transpose, Hermitian transpose, or adjoint matrix of an m-by-n matrix A with complex entries is the n-by-m matrix A* obtained from A by taking the transpose and then taking the complex conjugate of each entry. ...

Any induced norm satisfies the inequality

where ρ(A) is the spectral radius of A. Furthermore, we have In mathematics, the spectral radius of a matrix or a bounded linear operator is the supremum among the moduli of the elements in its spectrum, which is sometimes denoted by ρ(·). // Matrix Let λ1,...,λn be the (real or complex) eigenvalues of a matrix A. Then ρ(A) := max(|λi|) The spectral...

"Entrywise" norms

These vector norms treat a matrix as an vector, and use one of the familiar vector norms. Most entrywise norms are not (submultiplicative) matrix norms.

For example, using the p-norm for vectors, we get:

Frobenius norm

For p = 2, this is called the Frobenius norm. This norm can be defined in various ways:

where A^† denotes the conjugate transpose of A, σ_i 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. In mathematics, the conjugate transpose, Hermitian transpose, or adjoint matrix of an m-by-n matrix A with complex entries is the n-by-m matrix A* obtained from A by taking the transpose and then taking the complex conjugate of each entry. ... In mathematics, in particular functional analysis, singular values, or s-numbers of an bounded operator T acting on a Hilbert space are defined as the eigenvalues of (T*T)1/2. ... In linear algebra, the trace of an n-by-n square matrix A is defined to be the sum of the elements on the main diagonal (the diagonal from the upper left to the lower right) of A, i. ... In mathematics, an inner product space is a vector space with additional structure, an inner product (also called a scalar product), which allows us to introduce geometrical notions such as angles and lengths of vectors. ...

The Frobenius norm is submultiplicative and is very useful for numerical linear algebra. This norm is often more natural and more convenient than the induced norms. Numerical linear algebra is often at the heart of many engineering and computational science problems, such as image and signal processing, computational finance, materials science simulations, structural biology, datamining, and bioinformatics just to name a few. ...

Trace norm

The trace norm is defined as

Clearly, for all we have

Consistent norms

A matrix norm on is called consistent with a vector norm on Kⁿ and a vector norm on K^m if:

for all . All induced norms are consistent 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 other words, they are equivalent norms; they induce the same topology on the real or complex vector space. A Möbius strip, an object with only one surface and one edge; such shapes are an object of study in topology. ...

Moreover, when , then for any vector norm ||·||, there exists a unique positive number k such that k||A|| is a (submultiplicative) matrix norm.

A matrix norm ||·||_p is said to be minimal if there exists no other matrix norm ||·||_q satisfying ||·||_q≤||·||_p for all ||·||_q.

Examples of norm equivalence

For matrix the following inequalities hold:

• ^[1]
• ^[1]
• ^[1]
• ^[1]
• ^[1]

References

1. ^ ^{a b c d e} 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.
• L. Thomas, Norms and Condition Numbers of a Matrix [1]
• James W. Demmel, Applied Numerical Linear Algebra, section 1.7, published by SIAM, 1997.