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...

two-party systems, political polarization usually embodies the tension of its binary political ideologies and partisan identities. However, some political...

the norm is unique as a consequence of the polarization identity. In the real case, the polarization identity is given by any of the expressions: ⟨ x ,...

parallelogram identity holds is a Hilbert space, and the inner product is uniquely determined by the norm by the polarization identity. For real Hilbert...

variables Polarization identity, expresses an inner product in terms of its associated norm Polarization (Lie algebra) Polarization (physics), the ability...

sufficient condition for a norm to be defined by an inner product. Polarization identity ‖ x + y ‖ 2 = ‖ x ‖ 2 + ‖ y ‖ 2 + 2 Re ⁡ ⟨ x , y ⟩ {\displaystyle...

the origin, called the light cone in Minkowski space. Using the polarization identity the quadratic form is converted to a symmetric bilinear form called...

Political polarization is a prominent component of politics in the United States. Scholars distinguish between ideological polarization (differences between...

ISBN 978-1912475575. Christopher T. Stout. 2020. The Case for Identity Politics: Polarization, Demographic Change, and Racial Appeals. University of Virginia...

) {\displaystyle \epsilon _{\mu }(k)} is the polarization vector of the photon. Then the Ward identity reads: k μ M μ ( k ) = 0 {\displaystyle k_{\mu...

{\displaystyle H^{*}} satisfies the parallelogram law, which means that the polarization identity can be used to define a canonical inner product on H ∗ , {\displaystyle...

is the subject of non-commutative harmonic analysis. Due to the polarization identity, one can also apply Plancherel's theorem to the L 2 ( R ) {\displaystyle...

Polarization, or polarisation, is a property of transverse waves which specifies the geometrical orientation of the oscillations. In a transverse wave...

A linear map is unitary if it is surjective and isometric. (Use Polarization identity to show the only if part.) Antiunitary – Bijective antilinear map...

operators. For a complex Hilbert space, it is easy to verify by the polarization identity, that Strong Operator convergence implies Weak Operator convergence...

property of a material having a refractive index that depends on the polarization and propagation direction of light. These optically anisotropic materials...

{\displaystyle x} ) completely determines M {\displaystyle M} via the polarization identity; indeed, this remains true even if we allow M {\displaystyle M}...

the Hessian matrix is a covariant; see Invariant of a binary form Polarization identity, useful for rapid calculations involving Hessians. Jacobian matrix –...

inner product can be expressed in terms of the norm by using the polarization identity. On ℓ 2 , {\displaystyle \ell ^{2},} this inner product is the Euclidean...

In social psychology, group polarization refers to the tendency for a group to make decisions that are more extreme than the initial inclination of its...

{X}}^{\prime }} satisfies the parallelogram law, which means that the polarization identity can be used to define a canonical inner product on X ′ {\displaystyle...

\|A-B\|^{2}=(A-B)\cdot (A-B)=\|A\|^{2}+\|B\|^{2}-2(A\cdot B)\ } (polarization identity) by expansion. When A and B are normalized to unit length, ‖ A ‖...

This inner product can expressed in terms of the norm by using the polarization identity. On ℓ 2 , {\displaystyle \ell ^{2},} this is the Euclidean inner...

using the polarization identity. Greek mathematics Parallelogram law – Sides and diagonals have equal sums of squares Polarization identity – Formula...

{\displaystyle p=2.} If this identity is satisfied, the associated inner product is given by the polarization identity. In the case of real scalars,...

covariation may be written in terms of the quadratic variation by the polarization identity: [ X , Y ] t = 1 2 ( [ X + Y ] t − [ X ] t − [ Y ] t ) . {\displaystyle...

Hermitian, or self adjoint. The general claim can be argued using the polarization identity. That is no longer true in the real case. Fourier's law of heat...

In sociology, gender polarization is a concept created by American psychologist Sandra Bem which states that societies tend to define femininity and masculinity...

y ∈ Dom ⁡ A , {\displaystyle x,y\in \operatorname {Dom} A,} the polarization identity ⟨ A x , y ⟩ = 1 4 ( ⟨ A ( x + y ) , x + y ⟩ − ⟨ A ( x − y ) , x...

any quadratic form then the polarization of f {\displaystyle f} agrees with the conclusion of the polarization identity. A cubic example. Let f ( x ...