In
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, the operator norm measures the "size" of certain
linear operator
In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that pr ...
s by assigning each a
real number
In mathematics, a real number is a number that can be used to measure a continuous one- dimensional quantity such as a duration or temperature. Here, ''continuous'' means that pairs of values can have arbitrarily small differences. Every re ...
called its . Formally, it is a
norm defined on the space of
bounded linear operators between two given
normed vector spaces. Informally, the operator norm
of a linear map
is the maximum factor by which it "lengthens" vectors.
Introduction and definition
Given two normed vector spaces
and
(over the same base
field, either the
real number
In mathematics, a real number is a number that can be used to measure a continuous one- dimensional quantity such as a duration or temperature. Here, ''continuous'' means that pairs of values can have arbitrarily small differences. Every re ...
s
or the
complex number
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the for ...
s
), a
linear map
In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that p ...
is continuous
if and only if
In logic and related fields such as mathematics and philosophy, "if and only if" (often shortened as "iff") is paraphrased by the biconditional, a logical connective between statements. The biconditional is true in two cases, where either bo ...
there exists a real number
such that
The norm on the left is the one in
and the norm on the right is the one in
.
Intuitively, the continuous operator
never increases the length of any vector by more than a factor of
Thus the
image
An image or picture is a visual representation. An image can be Two-dimensional space, two-dimensional, such as a drawing, painting, or photograph, or Three-dimensional space, three-dimensional, such as a carving or sculpture. Images may be di ...
of a bounded set under a continuous operator is also bounded. Because of this property, the continuous linear operators are also known as
bounded operators.
In order to "measure the size" of
one can take the
infimum
In mathematics, the infimum (abbreviated inf; : infima) of a subset S of a partially ordered set P is the greatest element in P that is less than or equal to each element of S, if such an element exists. If the infimum of S exists, it is unique ...
of the numbers
such that the above inequality holds for all
This number represents the maximum scalar factor by which
"lengthens" vectors.
In other words, the "size" of
is measured by how much it "lengthens" vectors in the "biggest" case. So we define the operator norm of
as
The infimum is attained as the set of all such
is
closed,
nonempty, and
bounded from below.
It is important to bear in mind that this operator norm depends on the choice of norms for the normed vector spaces
and
.
Examples
Every real
-by-
matrix
Matrix (: matrices or matrixes) or MATRIX may refer to:
Science and mathematics
* Matrix (mathematics), a rectangular array of numbers, symbols or expressions
* Matrix (logic), part of a formula in prenex normal form
* Matrix (biology), the m ...
corresponds to a linear map from
to
Each pair of the plethora of (vector)
norms applicable to real vector spaces induces an operator norm for all
-by-
matrices of real numbers; these induced norms form a subset of
matrix norm
In the field of mathematics, norms are defined for elements within a vector space. Specifically, when the vector space comprises matrices, such norms are referred to as matrix norms. Matrix norms differ from vector norms in that they must also ...
s.
If we specifically choose the
Euclidean norm
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are ''Euclidean spaces'' ...
on both
and
then the matrix norm given to a matrix
is the
square root
In mathematics, a square root of a number is a number such that y^2 = x; in other words, a number whose ''square'' (the result of multiplying the number by itself, or y \cdot y) is . For example, 4 and −4 are square roots of 16 because 4 ...
of the largest
eigenvalue
In linear algebra, an eigenvector ( ) or characteristic vector is a vector that has its direction unchanged (or reversed) by a given linear transformation. More precisely, an eigenvector \mathbf v of a linear transformation T is scaled by a ...
of the matrix
(where
denotes the
conjugate transpose of
).
This is equivalent to assigning the largest
singular value of
Passing to a typical infinite-dimensional example, consider the
sequence space
In functional analysis and related areas of mathematics, a sequence space is a vector space whose elements are infinite sequences of real or complex numbers. Equivalently, it is a function space whose elements are functions from the natural num ...
which is an
L''p'' space, defined by
This can be viewed as an infinite-dimensional analogue of the
Euclidean space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are ''Euclidean spaces ...
Now consider a bounded sequence
The sequence
is an element of the space
with a norm given by
Define an operator
by pointwise multiplication:
The operator
is bounded with operator norm
This discussion extends directly to the case where
is replaced by a general
space with
and
replaced by
Equivalent definitions
Let
be a linear operator between normed spaces. The first four definitions are always equivalent, and if in addition
then they are all equivalent:
:
If
then the sets in the last two rows will be empty, and consequently their
supremum
In mathematics, the infimum (abbreviated inf; : infima) of a subset S of a partially ordered set P is the greatest element in P that is less than or equal to each element of S, if such an element exists. If the infimum of S exists, it is unique, ...
s over the set