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Formula relating the norm and the inner product in an inner product space
In linear algebra, 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
Polarization_identity
Vector space with generalized dot product
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
Inner_product_space
Type of vector space in math
parallelogram identity holds is a Hilbert space, and the inner product is uniquely determined by the norm by the polarization identity. For real Hilbert
Hilbert_space
Sides and diagonals have equal sums of squares
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_law
Divergence of political attitudes
two-party systems, political polarization usually embodies the tension of its binary political ideologies and partisan identities. However, some political
Political_polarization
Topics referred to by the same term
variables Polarization identity, expresses an inner product in terms of its associated norm Polarization (Lie algebra) Polarization (physics), the ability
Polarization
Mathematical function often applied to matrices
than 1 / 2 {\displaystyle 1/2} . The bound relies on using the polarization identity, in the linear as well as the nonlinear case, and therefore only
Logarithmic_norm
Mathematical description of spacetime used in relativity
occurs when one event is on the light cone of the other. Using the polarization identity the quadratic form is converted to a symmetric bilinear form called
Minkowski_spacetime
White supremacist interpretation of Christianity
likelihood of polarization continuing, thus resulting in two separate Aryan Israel religions. Jeffrey Kaplan argues that Christian Identity represents revolution
Christian_Identity
Property of waves that can oscillate with more than one orientation
Polarization, or polarisation, is a property of transverse waves which specifies the geometrical orientation of the oscillations. In a transverse wave
Polarization_(waves)
Politics based on one's identity
ISBN 978-1912475575. Christopher T. Stout. 2020. The Case for Identity Politics: Polarization, Demographic Change, and Racial Appeals. University of Virginia
Identity_politics
Political polarization is a prominent component of politics in the United States. Scholars distinguish between ideological polarization (differences between
Political polarization in the United States
Political_polarization_in_the_United_States
Construct for Hermitian matrices
{\displaystyle x} ) completely determines M {\displaystyle M} via the polarization identity; indeed, this remains true even if we allow M {\displaystyle M}
Rayleigh_quotient
Relation between distances of four points
unique and it can be defined in terms of the norm by using the polarization identity. Greek mathematics – Mathematics of Ancient Greece and the Mediterranean
Ptolemy's_inequality
Theorem in harmonic analysis
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
Plancherel_theorem
Surjective bounded operator on a Hilbert space preserving the inner product
A linear map is unitary if it is surjective and isometric. (Use Polarization identity to show the only if part.) Antiunitary – Bijective antilinear map
Unitary_operator
Technique for expressing a polynomial in simpler fashion by using more variables
any quadratic form then the polarization of f {\displaystyle f} agrees with the conclusion of the polarization identity. A cubic example. Let f ( x
Polarization of an algebraic form
Polarization_of_an_algebraic_form
Similarity measure for number sequences
\|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 ‖
Cosine_similarity
Locally convex topology on function spaces
operators. For a complex Hilbert space, it is easy to verify by the polarization identity, that Strong Operator convergence implies Weak Operator convergence
Strong_operator_topology
Matrix of second derivatives
the Hessian matrix is a covariant; see Invariant of a binary form Polarization identity, useful for rapid calculations involving Hessians. Jacobian matrix –
Hessian_matrix
Measurement on a normed vector space
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
Dual_norm
Refractive property of materials
property of a material having a refractive index that depends on the polarization and propagation direction of light. These optically anisotropic materials
Birefringence
Normed vector space that is complete
{\displaystyle p=2.} If this identity is satisfied, the associated inner product is given by the polarization identity. In the case of real scalars,
Banach_space
Theorem about the dual of a Hilbert space
{\displaystyle H^{*}} satisfies the parallelogram law, which means that the polarization identity can be used to define a canonical inner product on H ∗ , {\displaystyle
Riesz_representation_theorem
Quantity defined for a stochastic process
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
Quadratic_variation
Identity in abelian theories due to gauge invariance
) {\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
Ward–Takahashi_identity
Quantum explanation of electromagnetic polarization
Photon polarization is the quantum mechanical description of the classical polarized sinusoidal plane electromagnetic wave. An individual photon can be
Photon_polarization
Sociological concept by Sandra Bem
In sociology, gender polarization is a concept first described by American psychologist Sandra Bem which states that societies tend to define femininity
Gender_polarization
Probability distribution
2 σ 2 {\textstyle 2\sigma ^{2}} . This is a special case of the polarization identity. If X 1 {\textstyle X_{1}} , X 2 {\textstyle X_{2}} are two independent
Normal_distribution
Quadratic form for which there is a non-zero vector on which the form evaluates to zero
1, where the products represent the quadratic form. Through the polarization identity the quadratic form is related to a symmetric bilinear form B(u,
Isotropic_quadratic_form
Tendency of a group to make more extreme decisions than the inclinations of its members
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
Group_polarization
Length in a vector space
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
Norm_(mathematics)
Gauge boson self-energy due to interactions with virtual particles
quantum field theory, and specifically quantum electrodynamics, vacuum polarization describes a process in which a background electromagnetic field produces
Vacuum_polarization
Indigenous identity fraud is the practice of non-Indigenous people incorrectly claiming Indigenous identity. The Indigenous Chamber of Commercaes states
Indigenous identity fraud in Canada and the United States
Indigenous_identity_fraud_in_Canada_and_the_United_States
Generalization of the concept of directional derivative
h} and k . {\displaystyle k.} By virtue of the bilinearity, the polarization identity holds D 2 F ( u ) { h , k } = 1 2 d 2 F ( u ; h + k ) − d 2 F (
Gateaux_derivative
2020 book by Ezra Klein
in which he analyzes political polarization in the United States. Focusing in particular on the growing polarization between the major political parties
Why_We're_Polarized
Type of homogeneous polynomial of degree 2
{1}{2}}[Q(x+y)-Q(x)-Q(y)]~.\end{aligned}}} The latter formula (the polarization identity) arises from expanding Q ( x + y ) = B ( x + y , x + y ) . {\displaystyle
Definite_quadratic_form
Family of linear transformations
first formula automatically satisfies the second one as well; see Polarization identity). Finding the solution to the simpler problem is just a matter of
Lorentz_transformation
Type of algebras, possibly non associative
associated bilinear form B(x,y) constructed with the norm N and a polarization identity: B ( x , y ) = [ N ( x + y ) − N ( x ) − N ( y ) ] / 2. {\displaystyle
Composition_algebra
Conjugate homogeneous additive map
{X}}^{\prime }} satisfies the parallelogram law, which means that the polarization identity can be used to define a canonical inner product on X ′ {\displaystyle
Antilinear_map
Circle that passes through the vertices of a triangle
center u {\displaystyle \mathbf {u} } of the circle. Using the polarization identity, these equations reduce to the condition that the matrix [ | v |
Circumcircle
Property of a mathematical matrix
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
Definite_matrix
Concept in social psychology
1980s, social identity theory introduced the concept of a social identity as a way in which to explain intergroup behaviour. "Social identity theory explores
Social_identity_theory
In mathematics, a linear operator acting on inner product space
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
Positive_operator
Communication theory
The social identity model of deindividuation effects (or SIDE model) is a theory developed in social psychology and communication studies. SIDE explains
Social identity model of deindividuation effects
Social_identity_model_of_deindividuation_effects
Tensor product space endowed with a special inner product
N}{\|e_{n}\|}<\infty } . The latter describes a pre-inner product through the polarization identity, so take the closed span of such simple tensors modulo that inner
Tensor product of Hilbert spaces
Tensor_product_of_Hilbert_spaces
Situation that reinforces beliefs by repetition inside a closed system
amidst identity-driven controversies. As echo chambers build upon identity politics and emotion, they can contribute to political polarization and neotribalism
Echo_chamber_(media)
Optical phenomenon
Unpolarized light is light with a random, time-varying polarization. Natural light, like most other common sources of visible light, is produced independently
Unpolarized_light
Function spaces generalizing finite-dimensional p norm spaces
inner product can be expressed in terms of the norm by using the polarization identity. On ℓ 2 , {\displaystyle \ell ^{2},} it can be defined by ⟨ ( x
Lp_space
Laser cooling technique
Polarization gradient cooling (PG cooling), or Sisyphus cooling, is a technique in laser cooling of atoms by dampening the motion of the trapped particles
Polarization_gradient_cooling
Problem in linear algebra
respectively. Another way, connected with invariant theory is via the polarization identity for a symmetric tensor (Glynn 2010). The formula generalizes to
Computing_the_permanent
Weak topology on function spaces
same as the closure of that set in the SOT. It follows from the polarization identity that a net { T α } {\displaystyle \{T_{\alpha }\}} converges to
Weak_operator_topology
Algebra based on a vector space with a quadratic form
is the symmetric bilinear form associated with Q, via the polarization identity. Quadratic forms and Clifford algebras in characteristic 2 form
Clifford_algebra
Construction in functional analysis, useful to solve differential equations
\int _{\sigma (T)}g\,d\mu _{h}=\langle h,g(T)h\rangle .} Via the polarization identity, one can recover (since H is assumed to be complex) ⟨ k , g ( T
Decomposition of spectrum (functional analysis)
Decomposition_of_spectrum_(functional_analysis)
Biological process
Macrophage polarization is a process by which macrophages adopt different functional programs in response to the signals from their microenvironment. This
Macrophage_polarization
Thing in mathematics and theoretical physics
determined as B(x, y) = 1/4(Q(x + y) − Q(x − y)). See Polarization identity for variations of this identity. Though not mentioned in the source, we must exclude
Quasi-sphere
Polynomial with all terms of degree two
quadratic forms are equivalent to symmetric bilinear forms (by the polarization identities), but at 2 they are different concepts; this distinction is particularly
Quadratic_form
2023 book by Naomi Klein
filmmaker Naomi Klein. In it, Klein examines the 2020s climate of political polarization and conspiracy thinking, by contrasting Klein's worldview with that of
Doppelganger: A Trip into the Mirror World
Doppelganger:_A_Trip_into_the_Mirror_World
Set of functions used to represent the electronic wave function
functions are added to describe polarization of the electron density of the atom in molecules. These are called polarization functions. For example, while
Basis_set_(chemistry)
Geometrical figure in a Euclidean space
geometrical definition of a eutactic star, by the polarization identity. Furthermore, both Schläfli's identity and Hadwiger's theorem give the same value of
Eutactic_star
Equations of light transmission and reflection
that the waves were electric and magnetic fields. For the first time, polarization could be understood quantitatively, as Fresnel's equations correctly
Fresnel_equations
Type of matrix
{1}{2}}(d_{0i}^{2}+d_{0j}^{2}-d_{ij}^{2})} (this is known as the polarization identity). For a n×n matrix A, a sequence of points x 1 , x 2 , … , x n {\displaystyle
Euclidean_distance_matrix
K-theory of quadratic forms
an isomorphism modulo 2-torsion, and which corresponds to the polarization identities. The quadratic and the symmetric L-groups are 4-fold periodic (the
L-theory
Functional analysis concept
{\displaystyle m(T)=0} , then T = 0 {\displaystyle T=0} by the polarization identity, and this case is clear. Consider the function { f : H → R f ( x
Compact operator on Hilbert space
Compact_operator_on_Hilbert_space
History of research by Augustin-Jean Fresnel
contributions to several areas of physical optics, including to diffraction, polarization, and double refraction. The appreciation of Fresnel's reconstruction
Fresnel's_physical_optics
Virtual online communities
maladies. Social media has also received criticism as worsening political polarization and undermining democracy, exacerbated by platform capture by vested
Social_media
Type of social identity
members of a group. Identity issues are therefore central to understanding the "us versus them", "good versus bad" polarization in the relationships
Political_identity
Racialized classification of people
press has scoured every exit poll for the latest evidence of racial polarization, not just in terms of white and black, but black and brown as well. See
Black_people
Opposition to a disliked political party
positions of a disliked party. It has been claimed to be the cause of severe polarization in American politics. It has also been studied in the Canadian context
Negative_partisanship
Projective variety that is also an algebraic group
is a morphism of the underlying algebraic varieties that preserves the identity element for the group structure. An isogeny is a finite-to-one morphism
Abelian_variety
(2020). "Negotiating Identity with the Homeland: The Syriac Orthodox of North America in the Early Twentieth Century". From Polarization to Cohabitation in
Terms_for_Syriac_Christians
Phenomenon in political science
country have sharply diverging political views. It is a form of political polarization. Typically, urban areas exhibit more liberal, left-wing, secular, progressive
Urban–rural_political_divide
Ability to be regarded as having an identity one does not
ability of a person to be regarded as a member of an identity group or category, such as racial identity, ethnicity, caste, social class, sexual orientation
Passing_(sociology)
French optical physicist (1788–1827)
that light waves are purely transverse, Fresnel explained the nature of polarization. He then worked on double refraction. Fresnel had a lifelong battle with
Augustin-Jean_Fresnel
Type of optical device
The modulation may be imposed on the phase, frequency, amplitude, or polarization of the beam. Modulation bandwidths extending into the gigahertz range
Electro-optic_modulator
Political polarization is the bimodal distribution, meaning that two obvious peaks of opinion in the political sense. It can be observed through people's
Political polarization in Turkey
Political_polarization_in_Turkey
Equations describing classical electromagnetism
is proportional to the enclosed charge, including bound charge due to polarization of material. The coefficient of the proportion is the permittivity of
Maxwell's_equations
Commentators opposed to identity politics and political correctness
themselves as reasonable thinkers and reinforcing narratives of political polarization. According to Weiss, many IDW members have identified as atheist, including
Intellectual_dark_web
Physical model of propagating energy
multiple polarization signals can be combined (i.e. interfered) to form new states of polarization, which is known as parallel polarization state generation
Electromagnetic_radiation
German social media personality and soccer fan
showing the greatness of the country, amidst a backdrop of political polarization and declining domestic and international opinion of the US, especially
Freddy_(soccer_fan)
Matrices important in quantum mechanics and the study of spin
states of two polarization filters for horizontal/vertical polarization, 45 degree polarization (right/left), and circular polarization (right/left).
Pauli_matrices
Election to the 18th Legislative Assembly of the Indian state of West Bengal
Besides, we were wrong for not being able to understand that Hindu polarization would reach this level". Although he did not expect the results to be
2026 West Bengal Legislative Assembly election
2026_West_Bengal_Legislative_Assembly_election
Mathematical identity concerning matrices
Capelli identity and relative discrete series of line bundles over tube domains (PDF) Williamson, S. (1981), "Symmetry operators, polarizations, and a
Capelli's_identity
Cohort succeeding Generation Alpha
likely to grow during Generation Beta's time due to growing political polarization and growing indistinguishability between real and AI-generated material
Generation_Beta
Political party in France
nothing against Islam per se.' "Marine Le Pen, Islamophobia, and the polarization of French society". The ILR School | Cornell University. Archived from
National_Rally
Social networking service owned by Meta Platforms
study published at Frontiers Media in 2023 found that there was more polarization of the user-base on Facebook than even far-right social networks like
Political party in the United States
deem to be too moderate. According to sociologist Joe Feagin, political polarization by racially extremist Republicans as well as their increased attention
Republican Party (United States)
Republican_Party_(United_States)
American politician and attorney (born 1942)
Retrieved August 20, 2014. Wood, B. Dan; Jorden, Soren (2017). "Party polarization america war over two social contracts". Cambridge University Press. Archived
Mitch_McConnell
Penetrative sexual activity for reproduction or sexual pleasure
Naomi R. Cahn; June Carbone (2010). Red families v. blue families: legal polarization and the creation of culture. Oxford University Press US. p. 129. ISBN 978-0-19-537217-5
Sexual_intercourse
Physics phenomenon
Measurements of physical properties such as position, momentum, spin, and polarization performed on entangled particles can, in some cases, be found to be perfectly
Quantum_entanglement
Non-conventional expression of gender in humans
norms. A person can be gender nonconforming regardless of their gender identity; for example, they may be transgender, non-binary, or cisgender. Transgender
Gender_nonconformity
Concept in classical electromagnetism
causing an increase in the state of polarization, expressed as the polarization density P. A changing state of polarization is equivalent to a current. Both
Ampère's_circuital_law
Branch of physics
the polarization density, and n is the refractive index, which comes from the linear term in P. Note that one can normally use the vector identity ∇ ×
Nonlinear_optics
Political party in Slovenia
social and institutional reform, and cooperation beyond traditional polarization. After 2024 European parliament elections there was speculation, about
Prerod
Physical quantity in electromagnetism
from the slight motion of charges bound in atoms, called dielectric polarization. The idea was conceived by James Clerk Maxwell in his 1861 paper "On
Displacement_current_density
President of the United States (2017–2021; since 2025)
doi:10.1080/15564886.2019.1671284. Ross, Bertrall L. (July 1, 2024). "Polarization, Populism, and the Crisis of American Democracy". Annual Review of Law
Donald_Trump
Political party in the United States
increasing civil rights initiatives advocated by Northeastern liberals. The polarization grew stronger after Roosevelt died. Southern Democrats formed a key part
Democratic Party (United States)
Democratic_Party_(United_States)
Theory in social psychology
group cohesion, group polarization, and collective action. In subsequent years the theory, often as part of the social identity approach, has been applied
Self-categorization_theory
United States voting pattern
still often see close elections. Feelings of cultural and political polarization between red and blue states, which have gained increased media attention
Red_states_and_blue_states
2018 book by Jonah Goldberg
of the West: How the Rebirth of Tribalism, Populism, Nationalism, and Identity Politics Is Destroying American Democracy is a book by conservative columnist
Suicide of the West (Goldberg book)
Suicide_of_the_West_(Goldberg_book)
POLARIZATION IDENTITY
POLARIZATION IDENTITY
Girl/Female
Tamil
Higher, North the direction, Name of a start (Princess of Virata, pupil of Arjuna as Brihhannala (his disguised identity as the eunuch dance teacher during the Pandavas final year of exile).)
Girl/Female
Bengali, Gujarati, Hindu, Indian, Kannada, Malayalam, Marathi, Oriya, Punjabi, Sikh, Sindhi, Tamil, Telugu
Glories; Love; Identity; Pride
Girl/Female
Indian
Identity
Surname or Lastname
English (of Norman origin)
English (of Norman origin) : habitational name from a place in northern France, of which the identity is not clear. It is probably Sainville in Eure-et-Loire, so called from Old French saisne ‘Saxon’ + ville ‘settlement’.
Boy/Male
Muslim
Identity
Surname or Lastname
English
English : metonymic occupational name for a felt maker, from Old English felt ‘felt’.Said to be an Americanized or Germanized spelling of a Hungarian name, of uncertain identity.
Girl/Female
African, American, Arabic, Australian, Gujarati, Indian, Jain, Japanese, Muslim, Sanskrit, Swahili, Tamil
Name; One's Self; The Victorious; Named Child; Identity
Girl/Female
Muslim
Identity
Boy/Male
Arabic, Gujarati, Hindu, Indian, Kannada, Muslim
Identity
Girl/Female
Hindu
Higher, North the direction, Name of a start (Princess of Virata, pupil of Arjuna as Brihhannala (his disguised identity as the eunuch dance teacher during the Pandavas final year of exile).)
POLARIZATION IDENTITY
POLARIZATION IDENTITY
Boy/Male
Australian, British, Danish, English, French
Wearing a Mustache
Girl/Female
Tamil
Santhini | ஸஂதீநீÂ
Boy/Male
Indian
People with this Name Tend to be Very Inspired Intuitive and Creative
Female
English
 English form of Latin Clara, CLARE means "clear, bright." Compare with masculine Clare.
Boy/Male
Assamese, Bengali, Gujarati, Hindu, Indian, Kannada, Tamil
Shape
Boy/Male
Assamese, Bengali, Gujarati, Hindu, Indian, Kannada, Malayalam, Marathi, Oriya, Sanskrit, Sindhi, Telugu, Traditional
Lord of Light
Boy/Male
Indian
Moon Rays
Girl/Female
Australian, Danish, Finnish, French, German, Greek, Italian, Swedish
Mountain of the God; From Mount Olympus; Home of the Gods; Heavenly
Boy/Male
Bengali, Gujarati, Hindu, Indian, Kannada, Malayalam, Marathi, Telugu
Blue Mountain; The Nilgiris
Boy/Male
Indian, Sanskrit
Overlord of Heaven
POLARIZATION IDENTITY
POLARIZATION IDENTITY
POLARIZATION IDENTITY
POLARIZATION IDENTITY
POLARIZATION IDENTITY
n.
A translucent, gummy, amorphous substance, nearly tasteless and odorless, used as a substitute for gum, for sizing, etc., and obtained from starch by the action of heat, acids, or diastase. It is of somewhat variable composition, containing several carbohydrates which change easily to their respective varieties of sugar. It is so named from its rotating the plane of polarization to the right; -- called also British gum, Alsace gum, gommelin, leiocome, etc. See Achroodextrin, and Erythrodextrin.
n.
A sirupy variety of sugar, rarely obtained crystallized, occurring widely in honey, ripe fruits, etc., and hence called also fruit sugar. It is called levulose, because it rotates the plane of polarization to the left.
a.
Having two axes; as, biaxial polarization.
n.
Injury of a photographic picture caused by exposing it for too long a time to the sun's light in the camera; burning; excessive insolation.
n.
The act of depriving of polarity, or the result of such action; reduction to an unpolarized condition.
a.
Turning or twisting the plane of polarization towards the left, as levulose, levotartaric acid, etc.
n.
The part of a polariscope which receives the light after polarization, and exhibits its properties.
a.
Susceptible of polarization.
n.
The act of making popular, or of introducing among the people.
a.
Turning, or causing to turn, toward the right hand; esp., turning the plane of polarization of luminous rays toward the right hand; as, dextrorotatory crystals, sugars, etc. Cf. Levorotatory.
a.
Turning or rotating the plane of polarization towards the left; levogyrate, as levulose, left-handed quartz crystals, etc.
n.
The act of polarizing; the state of being polarized, or of having polarity.
n.
A peculiar affection or condition of the rays of light or heat, in consequence of which they exhibit different properties in different directions.
n.
A substance used to prevent polarization, as upon the negative plate of a voltaic battery.
n.
The art or process of measuring the polarization of light.
n.
An instrument for determining the amount of polarization of light, or the proportion of polarized light, in a partially polarized ray.
a.
Producing rotation of the plane of polarization; as, the rotatory power of bodies on light. See the Note under polarization.
n.
An effect produced upon the plates of a voltaic battery, or the electrodes in an electrolytic cell, by the deposition upon them of the gases liberated by the action of the current. It is chiefly due to the hydrogen, and results in an increase of the resistance, and the setting up of an opposing electro-motive force, both of which tend materially to weaken the current of the battery, or that passing through the cell.
v. t.
To free from polarization, as the negative plate of the voltaic battery.
a.
Turning the plane of polarization circularly or spirally to the right or left.