Polarization identity

Vectors involved in the polarization identity.

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 $\|x\| \,$ denote the norm of vector x and $\langle x, \ y \rangle \,$ the inner product of vectors x and y. Then the underlying theorem, attributed to Fréchet, von Neumann and Jordan, is stated as:^[1][2]

In a normed space (V, $\| \cdot \|$), if the parallelogram law holds, then there is an inner product on V such that $\|x\|^2 = \langle x,\ x\rangle$ for all $x \in V$.

Formula

The various forms given below are all related by the parallelogram law:

$2\|\textbf{u}\|^2 + 2\|\textbf{v}\|^2 = \|\textbf{u}+\textbf{v}\|^2 + \|\textbf{u}-\textbf{v}\|^2.$

The polarization identity can be generalized to various other contexts in abstract algebra, linear algebra, and functional analysis.

For vector spaces with real scalars

If V is a real vector space, then the inner product is defined by the polarization identity

$\langle x, \ y \rangle = \frac{1}{4} \left(\|x + y \|^2 - \|x-y\|^2 \right)\ \forall \ x,y \in V \ .$

For vector spaces with complex scalars

If V is a complex vector space the inner product is given by the polarization identity:

$\langle x, \ y \rangle = \frac{1}{4} \left(\|x + y \|^2 - \|x-y\|^2 + i\|x+iy\|^2 - i\|x-iy\|^2\right) \ \forall\ x,y \in V ;$

where i is the imaginary unit. Note that this defines an inner product which is linear in its first and semilinear in its second argument. To adjust for contrary definition, one needs to take the complex conjugate.

Multiple special cases for the Euclidean norm

A special case is an inner product given by the dot product, the so-called standard or Euclidean inner product. In this case, common forms of the identity include:

$\begin{array}{lr} \textbf{u}\cdot\textbf{v} = \displaystyle\frac{1}{2}\left(\|\textbf{u}+\textbf{v}\|^2 - \|\textbf{u}\|^2 - \|\textbf{v}\|^2\right),\quad & (1) \\[1.5em] \textbf{u}\cdot\textbf{v} = \displaystyle\frac{1}{2}\left(\|\textbf{u}\|^2 + \|\textbf{v}\|^2 - \|\textbf{u}-\textbf{v}\|^2 \right), & (2) \\[1.5em] \textbf{u}\cdot\textbf{v} = \displaystyle\frac{1}{4}\left(\|\textbf{u}+\textbf{v}\|^2 - \|\textbf{u}-\textbf{v}\|^2 \right). & (3) \end{array}$

Application to dot products

Relation to the law of cosines

The second form of the polarization identity can be written as

$\|\textbf{u}-\textbf{v}\|^2 = \|\textbf{u}\|^2 + \|\textbf{v}\|^2 - 2(\textbf{u}\cdot\textbf{v}).$

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,

$\textbf{u}\cdot\textbf{v} = \|\textbf{u}\|\,\|\textbf{v}\| \cos\theta,$

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 equation

$\|\textbf{v}\|^2 = \textbf{v} \cdot \textbf{v}.$

Then

$\begin{alignat}{2} \|\textbf{u}+\textbf{v}\|^2 &= (\textbf{u}+\textbf{v})\cdot(\textbf{u}+\textbf{v}) \\[3pt] &= (\textbf{u}\cdot\textbf{u}) + (\textbf{u}\cdot\textbf{v}) + (\textbf{v}\cdot\textbf{u}) + (\textbf{v}\cdot\textbf{v}) \\[3pt] &= \|\textbf{u}\|^2 + \|\textbf{v}\|^2 + 2(\textbf{u}\cdot\textbf{v}), \end{alignat}$

and similarly

$\|\textbf{u}-\textbf{v}\|^2 = \|\textbf{u}\|^2 + \|\textbf{v}\|^2 - 2(\textbf{u}\cdot\textbf{v}).$

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.)

Generalizations

Norms

In linear algebra, the polarization identity applies to any norm on a vector space defined in terms of an inner product by the equation

$\|v\| = \sqrt{\langle v, v \rangle}.$

As noted for the dot product case above, for real vectors u and v, an angle θ can be introduced using:^[3]

$\langle u,\ v \rangle = \|u\| \|v\| \cos \theta \ ; \ (-\pi < \theta \le \pi)\ ,$

which is acceptable by virtue of the Cauchy–Schwarz inequality:

$\langle u,\ v \rangle \le \|u\| \|v\| \ .$

This inequality insures that the magnitude of the above defined cosine ≤ 1. The choice of the cosine function ensures that when $\langle u,\ v \rangle = 0 \,$ (orthogonal vectors), the angle θ = π/2.

In this case, the identities become

$\begin{array}{l} \langle u, v \rangle = \frac{1}{2}\left(\|u+v\|^2 - \|u\|^2 - \|v\|^2\right), \\[3pt] \langle u, v \rangle = \frac{1}{2}\left(\|u\|^2 + \|v\|^2 - \|u-v\|^2\right), \\[3pt] \langle u, v \rangle = \frac{1}{4}\left(\|u+v\|^2 - \|u-v\|^2\right). \end{array}$

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.

Symmetric bilinear forms

The polarization identities are not restricted to inner products. If B is any symmetric bilinear form on a vector space, and Q is the quadratic form defined by

$Q(v) = B(v,v),\,\!$

then

$\begin{align} 2 B(u,v) &= Q(u+v) - Q(u) - Q(v), \\ 2 B(u,v) &= Q(u) + Q(v) - Q(u-v), \\ 4 B(u,v) &= Q(u+v) - Q(u-v). \end{align}$

The so-called symmetrization map generalizes the latter formula, replacing Q by a homogeneous 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.

Complex numbers

In linear algebra over the complex numbers, it is customary to use a sesquilinear inner product, with the property that $\langle v,u\rangle$ is the complex conjugate of $\langle u,v\rangle$. In this case the standard polarization identities only give the real part of the inner product:

$\begin{array}{l} \text{Re}\langle u, v \rangle = \frac{1}{2}\left(\|u+v\|^2 - \|u\|^2 - \|v\|^2\right), \\[3pt] \text{Re}\langle u, v \rangle = \frac{1}{2}\left(\|u\|^2 + \|v\|^2 - \|u-v\|^2\right), \\[3pt] \text{Re}\langle u, v \rangle = \frac{1}{4}\left(\|u+v\|^2 - \|u-v\|^2\right). \end{array}$

Using $\text{Im}\langle u, v \rangle =\text{Re}\langle u, -iv \rangle$ (holds with the convention that the inner product is linear in the second variable), the imaginary part of the inner product can be retrieved as follows:

$\begin{array}{l} \text{Im}\langle u, v \rangle = \frac{1}{2}\left(\|u-iv\|^2 - \|u\|^2 - \|v\|^2\right), \\[3pt] \text{Im}\langle u, v \rangle = \frac{1}{2}\left(\|u\|^2 + \|v\|^2 - \|u+iv\|^2\right), \\[3pt] \text{Im}\langle u, v \rangle = \frac{1}{4}\left(\|u-iv\|^2 - \|u+iv\|^2\right). \end{array}$

Homogeneous polynomials of higher degree

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.

The polarization identity can be stated in the following way:

$\langle u, v \rangle = 4^{-1} \sum_{k=0}^3 i^k\|u+i^k v\|^2.$

Notes and references

1. Philippe Blanchard, Erwin Brüning (2003). "Proposition 14.1.2 (Fréchet–von Neumann–Jordan)". Mathematical methods in physics: distributions, Hilbert space operators, and variational methods. Birkhäuser. p. 192. ISBN 0817642285.
2. Gerald Teschl (2009). "Theorem 0.19 (Jordan–von Neumann)". Mathematical methods in quantum mechanics: with applications to Schrödinger operators. American Mathematical Society Bookstore. p. 19. ISBN 0-8218-4660-4.
3. Francis Begnaud Hildebrand (1992). "Equation 66, the natural definition". Methods of applied mathematics (Reprint of Prentice-Hall 1965 2nd ed.). Courier Dover Publications. p. 24. ISBN 0-486-67002-3.