Polarization identity
Encyclopedia
In mathematics
, the polarization identity is any one of a family of formulas that express the inner product of two vectors in terms of the norm
of a normed vector space
. Let
denote the norm of vector x and 
the inner product of vectors x and y. Then the underlying theorem, attributed to Fréchet, von Neumann and Jordan
, is stated as:
(The notation “
” means “for all vectors x, y in vector space V”.)
The polarization identity can be generalized to various other contexts in abstract algebra
, linear algebra
, and functional analysis
.
.
where i = √(−1) .
, the so-called standard or Euclidean inner product. In this case, common forms of the identity include:
This is essentially a vector form of the law of cosines
for the triangle
formed by the vectors u, v, and u – v. In particular,
where θ is the angle between the vectors u and v.
Then
and similarly
Forms (1) and (2) of the polarization identity now follow by solving these equations for u · v, while form (3) follows from subtracting these two equations. (Adding these two equations together gives the parallelogram law.)
, the polarization identity applies to any norm
on a vector space
defined in terms of an inner product by the equation
As noted for the dot product case above, for real vectors u and v, an angle θ can be introduced using:
which is acceptable by virtue of the Cauchy–Schwarz inequality
:
This inequality insures that the magnitude of the above defined cosine ≤ 1. The choice of the cosine function ensures that when
(orthogonal vectors), the angle θ = π/2.
In this case, the identities become
Conversely, if a norm on a vector space satisfies the parallelogram law, then any one of the above identities can be used to define a compatible inner product. In functional analysis, introduction of an inner product norm like this often is used to make a Banach space
into a Hilbert space
.
on a vector space, and Q is the quadratic form
defined by
then
The so-called symmetrization map
generalizes the latter formula, replacing Q by a homogenous polynomial of degree k defined by Q(v)=B(v,...,v), where B is a symmetric k-linear map.
The formulas above even apply in the case where the field
of scalars
has characteristic
two, though the left-hand sides are all zero in this case. Consequently, in characteristic two there is no formula for a symmetric bilinear form in terms of a quadratic form, and they are in fact distinct notions, a fact which has important consequences in L-theory
; for brevity, in this context "symmetric bilinear forms" are often referred to as "symmetric forms".
These formulas also apply to bilinear forms on modules
over a commutative ring
, though again one can only solve for B(u, v) if 2 is invertible in the ring, and otherwise these are distinct notions. For example, over the integers, one distinguishes integral quadratic forms from integral symmetric forms, which are a narrower notion.
More generally, in the presence of a ring involution or where 2 is not invertible, one distinguishes ε-quadratic forms and ε-symmetric forms; a symmetric form defines a quadratic form, and the polarization identity (without a factor of 2) from a quadratic form to a symmetric form is called the "symmetrization map", and is not in general an isomorphism. This has historically been a subtle distinction: over the integers it was not until the 1950s that relation between "twos out" (integral quadratic form) and "twos in" (integral symmetric form) was understood - see discussion at integral quadratic form; and in the algebraization of surgery theory
, Mishchenko originally used symmetric L-groups, rather than the correct quadratic L-groups (as in Wall and Ranicki) - see discussion at L-theory
.
s, it is customary to use a sesquilinear
inner product, with the property that
is the complex conjugate
of
. In this case the standard polarization identities only give the real part of the inner product:
The imaginary part of the inner product can be retrieved as follows:
s (that is, algebraic forms) of arbitrary degree
, where it is known as the polarization formula, and is reviewed in greater detail in the article on the polarization of an algebraic form
.
The polarization identity can be stated in the following way:
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, the polarization identity is any one of a family of formulas that express the inner product of two vectors in terms of the norm
Norm (mathematics)
In linear algebra, functional analysis and related areas of mathematics, a norm is a function that assigns a strictly positive length or size to all vectors in a vector space, other than the zero vector...
of a normed vector space
Normed vector space
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. The following properties of "vector length" are crucial....
. Let
denote the norm of vector x and the inner product of vectors x and y. Then the underlying theorem, attributed to Fréchet, von Neumann and JordanPascual Jordan
-Further reading:...
, is stated as:
- In a normed space (V,
), if the parallelogram lawParallelogram lawIn mathematics, the simplest form of the parallelogram law belongs to elementary geometry. It states that the sum of the squares of the lengths of the four sides of a parallelogram equals the sum of the squares of the lengths of the two diagonals...
holds, then there is an inner product on V such that
(The notation “
” means “for all vectors x, y in vector space V”.)Formula
The various forms given below are all related by the parallelogram law:The polarization identity can be generalized to various other contexts in abstract algebra
Abstract algebra
Abstract algebra is the subject area of mathematics that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras...
, linear algebra
Linear algebra
Linear algebra is a branch of mathematics that studies vector spaces, also called linear spaces, along with linear functions that input one vector and output another. Such functions are called linear maps and can be represented by matrices if a basis is given. Thus matrix theory is often...
, and functional analysis
Functional analysis
Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure and the linear operators acting upon these spaces and respecting these structures in a suitable sense...
.
For vector spaces with real scalars
If V is a real vector space, then the inner product is defined by the polarization identity .For vector spaces with complex scalars
If V is a complex vector space the inner product is given by the polarization identity:where i = √(−1) .
Multiple special cases for the Euclidean norm
A special case is an inner product given by the dot productDot product
In mathematics, the dot product or scalar product is an algebraic operation that takes two equal-length sequences of numbers and returns a single number obtained by multiplying corresponding entries and then summing those products...
, the so-called standard or Euclidean inner product. In this case, common forms of the identity include:
Relation to the law of cosines
The second form of the polarization identity can be written asThis is essentially a vector form of the law of cosines
Law of cosines
In trigonometry, the law of cosines relates the lengths of the sides of a plane triangle to the cosine of one of its angles. Using notation as in Fig...
for the triangle
Triangle
A triangle is one of the basic shapes of geometry: a polygon with three corners or vertices and three sides or edges which are line segments. A triangle with vertices A, B, and C is denoted ....
formed by the vectors u, v, and u – v. In particular,
where θ is the angle between the vectors u and v.
Derivation
The basic relation between the norm and the dot product is given by the equationThen
and similarly
Forms (1) and (2) of the polarization identity now follow by solving these equations for u · v, while form (3) follows from subtracting these two equations. (Adding these two equations together gives the parallelogram law.)
Norms
In linear algebraLinear algebra
Linear algebra is a branch of mathematics that studies vector spaces, also called linear spaces, along with linear functions that input one vector and output another. Such functions are called linear maps and can be represented by matrices if a basis is given. Thus matrix theory is often...
, the polarization identity applies to any norm
Norm (mathematics)
In linear algebra, functional analysis and related areas of mathematics, a norm is a function that assigns a strictly positive length or size to all vectors in a vector space, other than the zero vector...
on a vector space
Vector space
A vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied by numbers, called scalars in this context. Scalars are often taken to be real numbers, but one may also consider vector spaces with scalar multiplication by complex...
defined in terms of an inner product by the equation
As noted for the dot product case above, for real vectors u and v, an angle θ can be introduced using:
which is acceptable by virtue of the Cauchy–Schwarz inequality
Cauchy–Schwarz inequality
In mathematics, the Cauchy–Schwarz inequality , is a useful inequality encountered in many different settings, such as linear algebra, analysis, probability theory, and other areas...
:
This inequality insures that the magnitude of the above defined cosine ≤ 1. The choice of the cosine function ensures that when
(orthogonal vectors), the angle θ = π/2.In this case, the identities become
Conversely, if a norm on a vector space satisfies the parallelogram law, then any one of the above identities can be used to define a compatible inner product. In functional analysis, introduction of an inner product norm like this often is used to make a Banach space
Banach space
In mathematics, Banach spaces is the name for complete normed vector spaces, one of the central objects of study in functional analysis. A complete normed vector space is a vector space V with a norm ||·|| such that every Cauchy sequence in V has a limit in V In mathematics, Banach spaces is the...
into a Hilbert space
Hilbert space
The mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space. It extends the methods of vector algebra and calculus from the two-dimensional Euclidean plane and three-dimensional space to spaces with any finite or infinite number of dimensions...
.
Symmetric bilinear forms
The polarization identities are not restricted to inner products. If B is any symmetric bilinear formSymmetric bilinear form
A symmetric bilinear form is a bilinear form on a vector space that is symmetric. Symmetric bilinear forms are of great importance in the study of orthogonal polarity and quadrics....
on a vector space, and Q is the quadratic form
Quadratic form
In mathematics, a quadratic form is a homogeneous polynomial of degree two in a number of variables. For example,4x^2 + 2xy - 3y^2\,\!is a quadratic form in the variables x and y....
defined by
then
The so-called symmetrization map
Homogeneous polynomial
In mathematics, a homogeneous polynomial is a polynomial whose monomials with nonzero coefficients all have thesame total degree. For example, x^5 + 2 x^3 y^2 + 9 x y^4 is a homogeneous polynomial...
generalizes the latter formula, replacing Q by a homogenous polynomial of degree k defined by Q(v)=B(v,...,v), where B is a symmetric k-linear map.
The formulas above even apply in the case where the field
Field (mathematics)
In abstract algebra, a field is a commutative ring whose nonzero elements form a group under multiplication. As such it is an algebraic structure with notions of addition, subtraction, multiplication, and division, satisfying certain axioms...
of scalars
Scalar (mathematics)
In linear algebra, real numbers are called scalars and relate to vectors in a vector space through the operation of scalar multiplication, in which a vector can be multiplied by a number to produce another vector....
has characteristic
Characteristic (algebra)
In mathematics, the characteristic of a ring R, often denoted char, is defined to be the smallest number of times one must use the ring's multiplicative identity element in a sum to get the additive identity element ; the ring is said to have characteristic zero if this repeated sum never reaches...
two, though the left-hand sides are all zero in this case. Consequently, in characteristic two there is no formula for a symmetric bilinear form in terms of a quadratic form, and they are in fact distinct notions, a fact which has important consequences in L-theory
L-theory
Algebraic L-theory is the K-theory of quadratic forms; the term was coined by C. T. C. Wall,with L being used as the letter after K. Algebraic L-theory, also known as 'hermitian K-theory',is important in surgery theory.-Definition:...
; for brevity, in this context "symmetric bilinear forms" are often referred to as "symmetric forms".
These formulas also apply to bilinear forms on modules
Module (mathematics)
In abstract algebra, the concept of a module over a ring is a generalization of the notion of vector space, wherein the corresponding scalars are allowed to lie in an arbitrary ring...
over a commutative ring
Commutative ring
In ring theory, a branch of abstract algebra, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra....
, though again one can only solve for B(u, v) if 2 is invertible in the ring, and otherwise these are distinct notions. For example, over the integers, one distinguishes integral quadratic forms from integral symmetric forms, which are a narrower notion.
More generally, in the presence of a ring involution or where 2 is not invertible, one distinguishes ε-quadratic forms and ε-symmetric forms; a symmetric form defines a quadratic form, and the polarization identity (without a factor of 2) from a quadratic form to a symmetric form is called the "symmetrization map", and is not in general an isomorphism. This has historically been a subtle distinction: over the integers it was not until the 1950s that relation between "twos out" (integral quadratic form) and "twos in" (integral symmetric form) was understood - see discussion at integral quadratic form; and in the algebraization of surgery theory
Surgery theory
In mathematics, specifically in geometric topology, surgery theory is a collection of techniques used to produce one manifold from another in a 'controlled' way, introduced by . Surgery refers to cutting out parts of the manifold and replacing it with a part of another manifold, matching up along...
, Mishchenko originally used symmetric L-groups, rather than the correct quadratic L-groups (as in Wall and Ranicki) - see discussion at L-theory
L-theory
Algebraic L-theory is the K-theory of quadratic forms; the term was coined by C. T. C. Wall,with L being used as the letter after K. Algebraic L-theory, also known as 'hermitian K-theory',is important in surgery theory.-Definition:...
.
Complex numbers
In linear algebra over the complex numberComplex number
A complex number is a number consisting of a real part and an imaginary part. Complex numbers extend the idea of the one-dimensional number line to the two-dimensional complex plane by using the number line for the real part and adding a vertical axis to plot the imaginary part...
s, it is customary to use a sesquilinear
Sesquilinear form
In mathematics, a sesquilinear form on a complex vector space V is a map V × V → C that is linear in one argument and antilinear in the other. The name originates from the numerical prefix sesqui- meaning "one and a half"...
inner product, with the property that
is the complex conjugateComplex conjugate
In mathematics, complex conjugates are a pair of complex numbers, both having the same real part, but with imaginary parts of equal magnitude and opposite signs...
of
. In this case the standard polarization identities only give the real part of the inner product:The imaginary part of the inner product can be retrieved as follows:
Homogeneous polynomials of higher degree
Finally, in any of these contexts these identities may be extended to homogeneous polynomialHomogeneous polynomial
In mathematics, a homogeneous polynomial is a polynomial whose monomials with nonzero coefficients all have thesame total degree. For example, x^5 + 2 x^3 y^2 + 9 x y^4 is a homogeneous polynomial...
s (that is, algebraic forms) of arbitrary degree
Degree (mathematics)
In mathematics, there are several meanings of degree depending on the subject.- Unit of angle :A degree , usually denoted by ° , is a measurement of a plane angle, representing 1⁄360 of a turn...
, where it is known as the polarization formula, and is reviewed in greater detail in the article on the polarization of an algebraic form
Polarization of an algebraic form
In mathematics, in particular in algebra, polarization is a technique for expressing a homogeneous polynomial in a simpler fashion by adjoining more variables...
.
The polarization identity can be stated in the following way: