Polarization identity
In linear algebra, a branch of 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.
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 satisfies the parallelogram law:
In fact, as observed by John von Neumann, 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 on a vector space induces a norm by the equationThe 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 numbers 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 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 numbers, the above formulas are not quite correct because they do not describe the imaginary part of the 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 in its argument.
They are related by the formula:
The real part of any inner product is a symmetric bilinear map that for any is always equal to:
It is always a symmetric map, meaning that
and it also satisfies:
which in plain English says that to move a factor of to the other argument, introduce a negative sign. These properties can be proven either from the properties of inner products directly or from properties of norms by using the polarization identity.
Let
Then
which proves that.
Additionally,
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 linear functional|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 lawholds, then there exists a unique inner product 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, 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 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 isometry between two Hilbert spaces thenthat 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 asThis is essentially a vector form of the law of cosines for the triangle formed by the vectors,, and.
In particular,
where is the angle between the vectors and.
The equation is numerically unstable if u and v are similar because of catastrophic cancellation and should be avoided for numeric computation.
Derivation
The basic relation between the norm and the dot product is given by the equationThen
and similarly
Forms and of the polarization identity now follow by solving these equations for, while form follows from subtracting these two equations.
Generalizations
Jordan–von Neumann theorems
The standard Jordan–von Neumann theorem, as stated previously, is that the if a norm satisfies the parallelogram law, then it can be induced by an inner product defined by the polarization identity. There are variants of the theorem.Define various senses of orthogonality:
- isosceles:
- Roberts’: for all scalar.
- Pythagorean:
- Birkhoff–James: for all scalar.
- The norm satisfies the parallelogram identity.
- for all unit vectors. That is, the norm satisfies the parallelogram identity for unit vectors.
- For any, the set of points equidistant to is flat, that is, an affine subspace.
- Orthogonality in either isosceles or Roberts’ sense is either additive or homogeneous on one variable.
- For every two-dimensional subspace, for every, there exists that is Roberts’ orthogonal to.
- Isosceles orthogonality implies Pythagorean orthogonality.
- Pythagorean orthogonality implies isosceles orthogonality.
- If are Pythagorean orthogonal, then so are.
- Birkhoff–James orthogonality is symmetric.
- If and are real, then.
- Any two-dimensional slice of the unit sphere is an ellipse, that is, parameterizable as, for some unit vectors.
The general case of the problem is open. When the space is parable finite-dimensional, the answer is yes. In other words, given a finite-dimensional normed vector space over the real or complex numbers, if any point on the unit sphere can be mapped to any other point by a linear isometry, then the norm is induced by an inner product.
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.
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 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 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" and "twos in" 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 – see discussion at L-theory.