In
linear algebra
Linear algebra is the branch of mathematics concerning linear equations such as:
:a_1x_1+\cdots +a_nx_n=b,
linear maps such as:
:(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n,
and their representations in vector spaces and through matrice ...
, a branch of
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, the polarization identity is any one of a family of formulas that express the
inner product
In mathematics, an inner product space (or, rarely, a Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation called an inner product. The inner product of two vectors in the space is a scalar, often ...
of two
vectors in terms of the
norm of a
normed vector space.
If a
norm arises from an inner product then the polarization identity can be used to express this inner product entirely in terms of the norm. The polarization identity shows that a norm can arise from at most one inner product; however, there exist norms that do not arise from any inner product.
The norm associated with any
inner product space
In mathematics, an inner product space (or, rarely, a Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation called an inner product. The inner product of two vectors in the space is a scalar, often ...
satisfies the
parallelogram law:
In fact, as observed by
John von Neumann
John von Neumann (; hu, Neumann János Lajos, ; December 28, 1903 – February 8, 1957) was a Hungarian-American mathematician, physicist, computer scientist, engineer and polymath. He was regarded as having perhaps the widest c ...
, the parallelogram law characterizes those norms that arise from inner products.
Given a
normed space , the parallelogram law holds for
if and only if there exists an inner product
on
such that
for all
in which case this inner product is uniquely determined by the norm via the polarization identity.
Polarization identities
Any
inner product
In mathematics, an inner product space (or, rarely, a Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation called an inner product. The inner product of two vectors in the space is a scalar, often ...
on a vector space induces a norm by the equation
The polarization identities reverse this relationship, recovering the inner product from the norm.
Every inner product satisfies:
Solving for
gives the formula
If the inner product is real then
and this formula becomes a polarization identity for real inner products.
Real vector spaces
If the vector space is over 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 distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...
s then the polarization identities are:
These various forms are all equivalent by the
parallelogram law:
This further implies that
class is not a
Hilbert space
In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise natural ...
whenever
, as the parallelogram law is not satisfied. For the sake of counterexample, consider
and
for any two disjoint subsets
of general domain
and compute the measure of both sets under parallelogram law.
Complex vector spaces
For vector spaces over 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 fo ...
s, the above formulas are not quite correct because they do not describe the
imaginary part of the (complex) inner product.
However, an analogous expression does ensure that both real and imaginary parts are retained.
The complex part of the inner product depends on whether it is
antilinear in the first or the second argument.
The notation
which is commonly used in physics will be assumed to be
antilinear in the argument while
which is commonly used in mathematics, will be assumed to be antilinear its the argument.
They are related by the formula:
The
real part of any inner product (no matter which argument is antilinear and no matter if it is real or complex) is a symmetric bilinear map that for any
is always equal to:
It is always a
symmetric map
In mathematics, symmetrization is a process that converts any function in n variables to a symmetric function in n variables.
Similarly, antisymmetrization converts any function in n variables into an antisymmetric function.
Two variables
Let S ...
, meaning that
and it also satisfies:
Thus
which in plain English says that to move a factor of
to the other argument, introduce a negative sign.
Let
Then
implies
and
Moreover,
which proves that
From
it follows that
and
so that
which proves that
Unlike its real part, the
imaginary part of a complex inner product depends on which argument is antilinear.
Antilinear in first argument
The polarization identities for the inner product
which is
antilinear in the argument, are
:
where
The second to last equality is similar to the formula
expressing a linear functional in terms of its real part:
Antilinear in second argument
The polarization identities for the inner product
which is
antilinear in the argument, follows from that of
by the relationship:
So for any
:
This expression can be phrased symmetrically as:
Summary of both cases
Thus if
denotes the real and imaginary parts of some inner product's value at the point
of its domain, then its imaginary part will be:
where the scalar
is always located in the same argument that the inner product is antilinear in.
Using
the above formula for the imaginary part becomes:
Reconstructing the inner product
In a normed space
if the
parallelogram law
holds, then there exists a unique
inner product
In mathematics, an inner product space (or, rarely, a Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation called an inner product. The inner product of two vectors in the space is a scalar, often ...
on
such that
for all
Another necessary and sufficient condition for there to exist an inner product that induces a given norm
is for the norm to satisfy
Ptolemy's inequality
In Euclidean geometry, Ptolemy's inequality relates the six distances determined by four points in the plane or in a higher-dimensional space. It states that, for any four points , , , and , the following inequality holds:
:\overline\cdot \over ...
, which is:
Applications and consequences
If
is a complex Hilbert space then
is real if and only if its imaginary part is
which happens if and only if
Similarly,
is (purely) imaginary if and only if
For example, from
it can be concluded that
is real and that
is purely imaginary.
Isometries
If
is a
linear
Linearity is the property of a mathematical relationship ('' function'') that can be graphically represented as a straight line. Linearity is closely related to '' proportionality''. Examples in physics include rectilinear motion, the linear ...
isometry
In mathematics, an isometry (or congruence, or congruent transformation) is a distance-preserving transformation between metric spaces, usually assumed to be bijective. The word isometry is derived from the Ancient Greek: ἴσος ''isos'' ...
between two Hilbert spaces (so
for all
) then
that is, linear isometries preserve inner products.
If
is instead an
antilinear isometry then
Relation to the law of cosines
The second form of the polarization identity can be written as
This is essentially a vector form of the
law of cosines for the
triangle
A triangle is a polygon with three edges and three vertices. It is one of the basic shapes in geometry. A triangle with vertices ''A'', ''B'', and ''C'' is denoted \triangle ABC.
In Euclidean geometry, any three points, when non- colline ...
formed by the vectors
and
In particular,
where
is the angle between the vectors
and
Derivation
The basic relation between the norm and the dot product is given by the equation
Then
and similarly
Forms (1) and (2) of the polarization identity now follow by solving these equations for
while form (3) follows from subtracting these two equations.
(Adding these two equations together gives the parallelogram law.)
Generalizations
Symmetric bilinear forms
The polarization identities are not restricted to inner products.
If
is any
symmetric bilinear form on a vector space, and
is the
quadratic form defined by
then
The so-called
symmetrization map generalizes the latter formula, replacing
by a homogeneous polynomial of degree
defined by
where
is a symmetric
-linear map.
[. See Keith Conrad (KCd)'s answer.]
The formulas above even apply in the case where the
field of
scalars
Scalar may refer to:
* Scalar (mathematics), an element of a field, which is used to define a vector space, usually the field of real numbers
* Scalar (physics), a physical quantity that can be described by a single element of a number field such ...
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
Broadly speaking, modularity is the degree to which a system's components may be separated and recombined, often with the benefit of flexibility and variety in use. The concept of modularity is used primarily to reduce complexity by breaking a s ...
over a
commutative ring, though again one can only solve for
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 forms, which are a narrower notion.
More generally, in the presence of a ring involution or where 2 is not invertible, one distinguishes
-quadratic forms and
-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
In mathematics, symmetrization is a process that converts any function in n variables to a symmetric function in n variables.
Similarly, antisymmetrization converts any function in n variables into an antisymmetric function.
Two variables
Let S ...
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 form) and "twos in" (integral form) was understood – see discussion at
integral quadratic form; and in the
algebraization of
surgery theory, Mishchenko originally used ''L''-groups, rather than the correct ''L''-groups (as in Wall and Ranicki) – see discussion at
L-theory.
Homogeneous polynomials of higher degree
Finally, in any of these contexts these identities may be extended to
homogeneous polynomial
In mathematics, a homogeneous polynomial, sometimes called quantic in older texts, is a polynomial whose nonzero terms all have the same degree. For example, x^5 + 2 x^3 y^2 + 9 x y^4 is a homogeneous polynomial of degree 5, in two variables; ...
s (that is,
algebraic forms) of arbitrary
degree, where it is known as the
polarization formula
In mathematics, in particular in algebra, polarization is a technique for expressing a homogeneous polynomial in a simpler fashion by adjoining more variables. Specifically, given a homogeneous polynomial, polarization produces a unique symmetri ...
, and is reviewed in greater detail in the article on the
polarization of an algebraic form.
See also
*
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*
*
*
Notes and references
Bibliography
*
*
*
{{Functional Analysis
Abstract algebra
Linear algebra
Functional analysis
Vectors (mathematics and physics)
Norms (mathematics)
Mathematical identities