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In mathematics, the operator norm is a means to measure the "size" of certain linear operators. Formally, it is a norm defined on the space of bounded linear operators between two given normed vector spaces. Euclid, Greek mathematician, 3rd century BC, known today as the father of geometry; shown here in a detail of The School of Athens by Raphael. ...
In mathematics, a linear transformation (also called linear operator or linear map) is a function between two vector spaces that respects the arithmetical operations addition and scalar multiplication defined on vector spaces, or, in other words, it preserves linear combinations. Definition and first consequences Formally, if V and W are...
In linear algebra, functional analysis and related areas of mathematics, a norm is a function which assigns a positive length or size to all vectors in a vector space, other than the zero vector. ...
In mathematics, the operator norm is a norm defined on the space of bounded operators between two Banach spaces. ...
Link titleIn 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. ...
Introduction and definition
Given two normed vector spaces V and W (over the same base field, either the real numbers R or the complex numbers C), a linear map A : V → W is continuous if and only if there exists a real number c such that In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication and division (except division by zero) may be performed, and the same rules hold which are familiar from the arithmetic of ordinary numbers. ...
In mathematics, the set of real numbers, denoted R, or in blackboard bold , is the set of all rational and irrational numbers. ...
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. ...
 (the norm on the left is the one in W, the norm on the right is the one in V). Intuitively, the continuous operator A never "lengthens" any vector more than by a factor of c. Because of this property, the continuous linear operators are also known as bounded operators. In order to "measure the size" of A, it then seems natural to take the smallest number c such that the above inequality holds for all v in V. In other words, we measure the "size" of A by how much it "lengthens" vectors in the worst case. So we define the operator norm of A as In mathematics, the operator norm is a norm defined on the space of bounded operators between two Banach spaces. ...
 (the minimum exists as the set of all such c is closed, nonempty, and bounded from below). In topology and related branches of mathematics, a closed set is a set whose complement is open. ...
In mathematics and more specifically set theory, the empty set is the unique set which contains no elements. ...
In mathematical analysis and related areas of mathematics, a set is called bounded, if it is, in a certain sense, of finite size. ...
Examples Every real m-by-n matrix yields a linear map from Rn to Rm. One can put several different norms on these spaces, as explained in the article on norms. Each such choice of norms gives rise to an operator norm and therefore yields a norm on the space of all m-by-n matrices. Examples can be found in the article on matrix norms. 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. ...
In linear algebra, functional analysis and related areas of mathematics, a norm is a function which assigns a positive length or size to all vectors in a vector space, other than the zero vector. ...
In mathematics, the term matrix norm can have two meanings: A vector norm on matrices, i. ...
If we specifically choose the Euclidean norm on both Rn and Rm, then we obtain the matrix norm which to a given a matrix A assigns the square root of the largest eigenvalue of the matrix A*A (where A* denotes the conjugate transpose of A). This is equivalent to assigning the largest singular value of A. In mathematics and astronomy, Euclidean space is a generalization of the 2- and 3-dimensional spaces studied by Euclid. ...
In mathematics, a square root of a number x is a number whose square (the result of multiplying the number by itself) is x. ...
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 mathematics, the conjugate transpose or adjoint 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. ...
Equivalent definitions One can show that the following definitions are all equivalent:  -
  
Properties The operator norm is indeed a norm on the space of all bounded operators between V and W. This means In mathematics, the operator norm is a norm defined on the space of bounded operators between two Banach spaces. ...
   Furthermore, we have the following important inequality  The operator norm is also compatible with the composition of operators: if V, W and X are three normed spaces over the same base field, and A : V → W and B: W → X are two bounded operators, then 
Operators on a Hilbert space Suppose H is a real or complex Hilbert space. If A : H → H is a bounded linear operator, then we have In mathematics, a Hilbert space is a real or complex vector space with a positive definite sesquilinear form, that is complete under its norm. ...
 and  where A* denotes the adjoint operator of A (which in finite dimensional situations corresponds to the conjugate transpose of the matrix A). In mathematics, the conjugate transpose or adjoint 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, the conjugate transpose or adjoint 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 general, the spectral radius of A is bounded above by the operator norm of A: 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 Ï(·). // Matrix Let λ1,...,λn be the (real or complex) eigenvalues of a matrix A. Then Ï(A) := max(|λi|) The spectral...
 If we have a Hermitian operator H, then using the spectral theorem we can show that On a finite-dimensional inner product space, a self-adjoint operator is one that is its own adjoint, or, equivalently, one whose matrix is Hermitian, where a Hermitian matrix is one which is equal to its own conjugate transpose. ...
In mathematics, particularly linear algebra and functional analysis, the spectral theorem is any of a number of results about linear operators or about matrices. ...
 This formula can sometimes be used to compute the operator norm of a given bounded operator A: define the Hermitian operator H = A*A, determine its spectral radius, and take the square root to obtain the operator norm of A. In mathematics, a square root of a number x is a number whose square (the result of multiplying the number by itself) is x. ...
The set of all bounded operators on a Hilbert space, together with the operator norm and the adjoint operation, yields a C*-algebra. C*-algebras are an important area of research in functional analysis. ...
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