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.
The norm associated with any inner product space satisfies the parallelogram law:
In fact, as observed by John von Neumann,[1] the parallelogram law characterizes those norms that arise from inner products.
in which case this inner product is uniquely determined by the norm via the polarization identity.
[2][3] Any inner product on a vector space induces a norm by the equation
The polarization identities reverse this relationship, recovering the inner product from the norm.
If the vector space is over the real numbers then the polarization identities are:[4]
These various forms are all equivalent by the parallelogram law:[proof 1]
and compute the measure of both sets under parallelogram law.
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.
which is commonly used in physics will be assumed to be antilinear in the first argument while
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
The second to last equality is similar to the formula expressing a linear functional
denotes the real and imaginary parts of some inner product's value at the point
{\displaystyle I(x,y)~=~{\begin{cases}~R({\color {red}i}x,y)&\qquad {\text{ if antilinear in the }}{\color {red}1}{\text{st argument}}\\~R(x,{\color {blue}i}y)&\qquad {\text{ if antilinear in the }}{\color {blue}2}{\text{nd argument}}\\\end{cases}}}
[4][1] We will only give the real case here; the proof for complex vector spaces is analogous.
By the above formulas, if the norm is described by an inner product (as we hope), then it must satisfy
Explicitly, the following will be shown: (This axiomatization omits positivity, which is implied by (1) and the fact that
But the latter claim can be verified by subtracting the following two further applications of the parallelogram identity:
Another necessary and sufficient condition for there to exist an inner product that induces a given norm
The basic relation between the norm and the dot product is given by the equation
Forms (1) and (2) of the polarization identity now follow by solving these equations for
(Adding these two equations together gives the parallelogram law.)
is any symmetric bilinear form on a vector space, and
The so-called symmetrization map generalizes the latter formula, replacing
These formulas also apply to bilinear forms on modules over a commutative ring, though again one can only solve for
More generally, in the presence of a ring involution or where 2 is not invertible, one distinguishes
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.
Finally, in any of these contexts these identities may be extended to homogeneous polynomials (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.
