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Vector on which a quadratic form is zero
In mathematics, given a vector space X with an associated quadratic form q, written (X, q), a null vector or isotropic vector is a non-zero element x
Null_vector
Mathematical description of spacetime used in relativity
null vectors, called a null basis. Vector fields are called timelike, spacelike, or null if the associated vectors are timelike, spacelike, or null at
Minkowski_spacetime
Mathematical representation of absence of a value
set" (e.g., null set) to "having a value of zero" (e.g., null vector). In a vector space, the null vector is the neutral element of vector addition; depending
Null_(mathematics)
Type of hypersurface
pseudo-Riemannian geometry, a null hypersurface is a hypersurface whose normal vector at every point is a null vector (has zero length with respect to
Null_hypersurface
Mathematics of general relativity
some scalar and vector quantities constructed from arbitrary timelike or null vectors and the matter tensor. First, a unit timelike vector field X → {\displaystyle
Energy_condition
In mathematics, vector subspace
homogeneous linear parametric equations, the span of a collection of vectors, and the null space, column space, and row space of a matrix. Geometrically (especially
Linear_subspace
Causal relationships between points in a manifold
tangent vector is past-directed. These definitions only apply to causal (chronological or null) curves because only timelike or null tangent vectors can be
Causal_structure
Vectors mapped to 0 by a linear map
a linear map, also known as the null space or nullspace, is the part of the domain which is mapped to the zero vector of the co-domain; the kernel is
Kernel_(linear_algebra)
Concept used in black hole solutions
trapped null surface is often used interchangeably. However, when discussing causal horizons, trapped null surfaces are defined as only null vector fields
Trapped_surface
Notation in general relativity
observables. In the case of the NP formalism, the vector basis chosen is a null tetrad: a set of four null vectors—two real, and a complex-conjugate pair. The
Newman–Penrose_formalism
Geometric object that has length and direction
physics, and engineering, a Euclidean vector or simply a vector (sometimes called a geometric vector or spatial vector) is a geometric object that has magnitude
Euclidean_vector
Vector in relativity
In special relativity, a four-vector (or 4-vector, sometimes Lorentz vector) is an element of a four-dimensional vector space object with four components
Four-vector
Space in mathematics and theoretical physics
0, so that ei + ej is a null vector. In a pseudo-Euclidean space with k < n, unlike in a Euclidean space, there exist vectors with negative scalar square
Pseudo-Euclidean_space
Topics referred to by the same term
Switched-mode power supply 0 vector, or null vector, a vector where all components are zero 0 vector space; see Examples of vector spaces 0-velocity surface
0V
Formulation in general relativity
pair of real null vectors and { m a , m ¯ a } {\displaystyle \{m^{a},{\bar {m}}^{a}\}} is a pair of complex null vectors. These tetrad vectors respect the
Construction of a complex null tetrad
Construction_of_a_complex_null_tetrad
Coordinate system
coordinate vector field dual to the covector field d v {\displaystyle dv} , is a null vector field. Indeed, geometrically speaking, it is a null geodesic
Brinkmann_coordinates
Linear error-correcting code
rows are used to compute the syndrome vector at the receiving end and if the syndrome vector is the null vector (all zeros) then the received word is
Hamming(7,4)
Associative algebra generalizing the Virasoro algebra
called a null state or null vector. A degenerate representation is obtained by setting one or more null vectors to zero. Setting all the null vectors to zero
W-algebra
Generalization of perpendicularity
. Depending on the bilinear form, the vector space may contain null vectors, non-zero self-orthogonal vectors, in which case perpendicularity is replaced
Orthogonality_(mathematics)
Matrix decomposition
{\displaystyle \mathbf {A} } 's null space and is sometimes called a (right) null vector of A {\displaystyle \mathbf {A} } . The vector x {\displaystyle \mathbf
Singular_value_decomposition
Euclidean geometry without distance and angles
called a null-vector; then the parallel transport in question is such that it carries any null-vector at one point into the position of a null-vector at a
Affine_geometry
Concept in linear algebra
The vector projection (also known as the vector component or vector resolution) of a vector a on (or onto) a non-zero vector b is the orthogonal projection
Vector_projection
Motion-independent mass, equals total mass when at rest
the speed of light squared. Systems whose four-momentum is a null vector, a light-like vector within the context of Minkowski space (for example, a single
Invariant_mass
Lie group of Lorentz transformations
by using the complex projective vector to construct a null-vector. If ξ {\displaystyle \xi } is a CP1 projective vector, it can be tensored with its Hermitian
Lorentz_group
Quadratic form for which there is a non-zero vector on which the form evaluates to zero
non-zero vector v in V is said to be isotropic if q(v) = 0. A quadratic form is isotropic if and only if there exists a non-zero isotropic vector (or null vector)
Isotropic_quadratic_form
is minimum; at that point the antenna's null is pointed along the bearing line to the transmitter. Null vector Carr, Joseph J. (1999). Practical radio
Null_(radio)
Concept in general relativity
states that any Lorentzian manifold which admits a covariantly constant null vector field k {\displaystyle k} is called a pp-wave spacetime. That is, the
Pp-wave_spacetime
Mathematical description of quantum state
state as well. (Whether the null vector counts as a valid state ("no system present") is a matter of definition. The null vector does not at any rate describe
Wave_function
Algebraic structure designed for geometry
along a non-null vector m {\displaystyle m} of the product of vectors as the reflection of every vector in the product along the same vector, we get for
Geometric_algebra
Type of algebras, possibly non associative
existence of a non-zero v in A such that N(v) = 0, called a null vector. When x is not a null vector, the multiplicative inverse of x is x ∗ N ( x ) {\textstyle
Composition_algebra
Vectors whose linear combinations are nonzero
a common plane. The three vectors define a three-dimensional space. The vectors o → {\displaystyle {\vec {o}}} (null vector, whose components are equal
Linear_independence
Length in a vector space
composition algebras does not share the usual properties of a norm since null vectors are allowed. A composition algebra ( A , ∗ , N ) {\displaystyle (A,{}^{*}
Norm_(mathematics)
Concept in mathematical physics
mathematical physics, a null dust solution (sometimes called a null fluid) is a Lorentzian manifold in which the Einstein tensor is null. Such a spacetime can
Null_dust_solution
Type of geometric algebra
resultant space of a map from points in an n-dimensional base space Rp,q to null vectors in Rp+1,q+1. This allows operations on the base space, including reflections
Conformal_geometric_algebra
Most widely known generalized inverse of a matrix
pseudoinverse of the null (all zero) vector is the transposed null vector. The pseudoinverse of a non-null vector is the conjugate transposed vector divided by
Moore–Penrose_inverse
are determined so to best approximate, in a least squares sense, the null vector. The newly determined coefficients are then used to extrapolate the function
DIIS
Vector spaces associated to a matrix
rank–nullity theorem. The null space of matrix A is the set of all vectors x for which Ax = 0. The product of the matrix A and the vector x can be written in
Row_and_column_spaces
Similarity measure for number sequences
the Euclidean distance between the vectors which were normalized to unit sum of squared values within them. Null distribution: For data which can be
Cosine_similarity
Uniformity in all orientations
be isotropic if there is a non-zero vector v such that q(v) = 0; such a v is an isotropic vector or null vector. In complex geometry, a line through
Isotropy
Algebra describing 2D conformal symmetry
module is irreducible if and only if it has no singular vectors. A singular vector or null vector of a highest weight representation is a state that is
Virasoro_algebra
Theory of interwoven space and time by Albert Einstein
the vectors are referred to as timelike, spacelike or null (lightlike). Note that a null vector is not the same as a zero vector. A null vector is one
Special_relativity
Classification used in differential geometry and general relativity
eigenbivectors are associated with certain null vectors in the original spacetime, which are called the principal null directions (at a given event). The relevant
Petrov_classification
subspace Null space, nullity Rank–nullity theorem Nullity theorem Dual space Linear function Linear functional Category of vector spaces Topological vector space
Outline_of_linear_algebra
Setting of relativistic physics in geometric algebra
the octonionic product as a Fano plane. A nonzero vector a {\displaystyle a} is a null vector (degree 2 nilpotent) if a 2 = 0 {\displaystyle a^{2}=0}
Spacetime_algebra
Vector describing a wave; often its propagation direction
In physics, a wave vector (or wavevector) is a vector used in describing a wave, with a typical unit being cycle per metre. It has a magnitude and direction
Wave_vector
Conformal field theory on a 2D spacetime
) = 0 {\displaystyle L_{-1}V_{1,1}(z)=0} , due to the presence of a null vector in the corresponding degenerate representation. An N {\displaystyle N}
Two-dimensional conformal field theory
Two-dimensional_conformal_field_theory
Point of reference in Euclidean space
zero. Coordinate frame Distance from a point to a plane Null vector, an analogous point of a vector space Pointed space, a topological space with a distinguished
Origin_(mathematics)
Real square matrix whose columns and rows are orthogonal unit vectors
1 transpositions. A Householder reflection is constructed from a non-null vector v as Q = I − 2 v v T v T v . {\displaystyle Q=I-2{\frac {{\mathbf {v}
Orthogonal_matrix
Open-source distributed vector database
Role-based access control Multi-vector and hybrid search Milvus 3.0 introduces the following features: Snapshots Nullable vector fields Evaluation rollbacks
Milvus_(vector_database)
Position that there is no relationship between two phenomena
The null hypothesis (often denoted H 0 {\textstyle H_{0}} ) is the claim in scientific research that the effect being studied does not exist. The null hypothesis
Null_hypothesis
1995 electronic game
configuration to be solvable (deriving the null vector from the original configuration) it must be orthogonal to the two vectors N1 and N2 below (pictured as a 5×5
Lights_Out_(game)
Various meanings of the terms
. Depending on the bilinear form, the vector space may contain null vectors, non-zero self-orthogonal vectors, in which case perpendicularity is replaced
Orthogonality
Polynomial with all terms of degree two
= x T A x . {\displaystyle q(x)=x^{\mathsf {T}}Ax.} A vector v = (x1, ..., xn) is a null vector if q(v) = 0. Two n-ary quadratic forms φ and ψ over K
Quadratic_form
with column j of B. null vector 1. Another term for an isotropic vector. 2. Another term for a zero vector. orthogonality Two vectors u and v are orthogonal
Glossary_of_linear_algebra
Object in geometric algebra
unit vectors. Reflections along a vector in geometric algebra may be represented as (minus) sandwiching a multivector M between a non-null vector v perpendicular
Rotor_(mathematics)
Generators of the Clifford algebra for relativistic quantum mechanics
the action of a set of orthogonal basis vectors for contravariant vectors in Minkowski space, the column vectors on which the matrices act become a space
Gamma_matrices
Upper-half plane model of hyperbolic non-Euclidean geometry
is obtained by orthographic projection in a direction parallel to a null vector, which can also be thought of as a kind of stereographic projection centered
Poincaré_half-plane_model
Alternative of event horizon first suggested by Stephen Hawking
\nu }} denotes the Ricci tensor and ℓ μ {\displaystyle \ell ^{\mu }} a null vector) is satisfied, then the apparent horizon is located inside the event
Apparent_horizon
Method in numerical analysis
coefficient of the null vector in the general solution of the Newton system (particular solution plus an arbitrary multiple of the null vector). This method
Numerical_continuation
In-band data value that must be handled specially by computer code
returning null pointers and null options. import std; using std::nullopt; using std::optional; using std::string; using std::string_view; using std::vector; struct
Sentinel_value
exist two future-directed null vector fields orthogonal to the surface, one outgoing and one ingoing. The corresponding null expansions θ + {\displaystyle
Hawking_energy
Four-vector
}}} is a unit vector in the direction of the photon's motion. The four-frequency of a photon is always a future-pointing and null vector. An observer moving
Four-frequency
Difference between the dimensions of mathematical object and a sub-object
constraints, exceeds N (in the linear algebra case, there is always a trivial, null vector solution, which is therefore discounted). The second is a matter of geometry
Codimension
Exact solution for the Einstein field equations
by Kerr and Schild in 1965. Notice that k is a unit 3-vector, making the 4-vector a null vector, with respect to both g and η. Here M is the constant
Kerr_metric
Vector satisfying some of the criteria of an eigenvector
eigenvector of an n × n {\displaystyle n\times n} matrix A {\displaystyle A} is a vector which satisfies certain criteria which are more relaxed than those for an
Generalized_eigenvector
Topic in special relativity
subgroup in question can be described as follows: The stabilizer of a null vector is the special Euclidean group SE(2), which contains T(2) as the subgroup
Very_special_relativity
Mathematical function
isotropic quadratic form so that A {\displaystyle A} has at least one null vector, contrary to the separation of points required for the usual norm discussed
Seminorm
Exact spherically symmetric solution in GR
l_{a}=-\partial _{a}u} and l a = g a b l b {\displaystyle l^{a}=g^{ab}l_{b}} are null (co)vectors (c.f. Box A below). Thus, T a b {\displaystyle T_{ab}} is a "pure radiation
Vaidya_metric
note that ∂u, ∂v are null vector fields. Therefore, ∂u + ∂v is a timelike vector field, while ∂u − ∂v, ∂x, ∂y are spacelike vector fields. To define the
Monochromatic electromagnetic plane wave
Monochromatic_electromagnetic_plane_wave
Set of integral curves of a vector field
Congruences generated by nowhere vanishing timelike, null, or spacelike vector fields are called timelike, null, or spacelike respectively. A congruence is called
Congruence (general relativity)
Congruence_(general_relativity)
Concept in algebraic geometry
(non-Hausdorff). (This is the case in our example – the null orbit is not open because any neighborhood of the null vector contains points in all other orbits, so in
Geometric_invariant_theory
Coordinates to capture characteristics of rotating frames of reference
{\displaystyle {\vec {e}}_{0}} is a timelike unit vector field while the others are spacelike unit vector fields; at each event, all four are mutually orthogonal
Born_coordinates
Method of statistical inference
statement about the sample). The test compares two hypotheses: a default "null" hypothesis (denoted H0) and its negation, the alternative hypothesis (H1)
Statistical_hypothesis_test
Model of the extended complex plane plus a point at infinity
equivalence classes of non-null vectors in the complex vector space C 2 {\displaystyle \mathbf {C} ^{2}} : two non-null vectors ( w , z ) {\displaystyle
Riemann_sphere
Differentiable manifold with nondegenerate metric tensor
positive-definite metrics, an indefinite signature allows tangent vectors to be classified into timelike, null or spacelike. With a signature of (p, 1) or (1, q), the
Pseudo-Riemannian_manifold
Mathematical operation on vectors in 3D space
M\mathbf {a} } The cross product of two vectors lies in the null space of the 2 × 3 matrix with the vectors as rows: a × b ∈ N S ( [ a b ] ) . {\displaystyle
Cross_product
Line along which a quadratic form applied to any two points' displacement is zero
quadratic forms, an isotropic line or null line is a line for which the quadratic form applied to the displacement vector between any pair of its points is
Isotropic_line
Mathematical concept
In mathematics, a symplectic vector space is a vector space V {\displaystyle V} over a field F {\displaystyle F} (for example the real numbers R {\displaystyle
Symplectic_vector_space
Theorem in general relativity
family of geodesic light-like curves. That is tangent vector field l a {\displaystyle l^{a}} is null and geodesic: l a l a = 0 {\displaystyle l_{a}l^{a}=0}
Goldberg–Sachs_theorem
Data structure
the vector string instructions to the IBM z13 in 2015. FreeBSD developer Poul-Henning Kamp, writing in ACM Queue, referred to the victory of null-terminated
Null-terminated_string
eigenvector r is a null vector (i.e. η(r,r) = 0), or the eigenvalue λ is zero, or both. A 1-dimensional subspace generated by a null eigenvector is called
Classification of electromagnetic fields
Classification_of_electromagnetic_fields
Distance from zero to a number
composition algebra may be a quadratic form that is not definite and has null vectors. However, as in the case of division algebras, when an element x has
Absolute_value
Euclidean space surface
\\r^{2}\end{pmatrix}}\ .} For t = 0 {\displaystyle t=0} the normal vector is the null vector. Hence the directrix consists of singular points. The directrix
Generalized_helicoid
Infinite cardinal number
ℵ 0 {\displaystyle \aleph _{0}} (read aleph-nought, aleph-zero, or aleph-null); the next larger cardinality of a well-ordered set is ℵ 1 , {\displaystyle
Aleph_number
Mathematical function, in linear algebra
subspace of X, called the null space of Λ {\displaystyle \Lambda } Rudin 1991, p. 14. Suppose now that X and Y are vector spaces over the same scalar
Linear_map
Type of Kac–Moody algebras
representations unless c acts by 0; this follows from the fact that the null vectors of a finite-dimensional Verma module are necessarily zero; whereas those
Affine_Lie_algebra
Line or vector perpendicular to a curve or a surface
linearly independent vectors pointing along the hyperplane, a normal to the hyperplane is any vector n {\displaystyle \mathbf {n} } in the null space of the matrix
Normal_(geometry)
Reals with an extra square root of +1 adjoined
\neq 0} ). Numbers of the form x ± j x have no inverse and are called null vectors. The multiplicative inverse of an invertible element is given by z −
Split-complex_number
Surname list
Minkowski distance Minkowski functional Minkowski inequality Minkowski space Null vector (Minkowski space) Minkowski plane Minkowski's theorem Minkowski's question
Minkowski
then X can be rescaled to a parallel vector field. In particular, non-parallel recurrent vector fields are null vector fields. Another example appears in
Recurrent_tensor
Coordinate system in special relativity
light-cone coordinate system, two of the coordinates are with respect to null vectors and all the other coordinates are spatial. The former can be denoted
Light-cone_coordinates
Generalization of the one-dimensional normal distribution to higher dimensions
normal distribution to higher dimensions. One definition is that a random vector is said to be k-variate normally distributed if every linear combination
Multivariate normal distribution
Multivariate_normal_distribution
Unique string of length zero
formal language theory, the empty string, also known as the empty word or null string, is the unique string of length zero. Formally, a string is a finite
Empty_string
Curve whose normals converge asymptotically
generating line of the cone " ) The normal of the cutting plane is a null vector in three-dimensional Minkowski space. If the metric is normalized to
Horocycle
Functions of complex quaternions
also gives a solution. Next find log(X) where X = a + N and N is a null non-zero vector quaternion. As easily verified log ( a + N ) = log ( a ) + 1
Biquaternion_functions
Property of geometry, also used to generalize the notion of "distance" in metric spaces
reversed form of the inequality holds if both vectors lie in the past light cone, and if one or both are null vectors. The result holds in n + 1 {\displaystyle
Triangle_inequality
Geometrical construct in general relativity
Mathematically a Killing horizon is a null hypersurface defined by the vanishing of the norm of a Killing vector field (both are named after Wilhelm Killing)
Killing_horizon
Geospatial vector data format
The shapefile format is a geospatial vector data format for geographic information system (GIS) software. It is developed and regulated by Esri as a mostly
Shapefile
Statistical method for handling multiple comparisons
discovery rate (FDR) is a method of conceptualizing the rate of type I errors in null hypothesis testing when conducting multiple comparisons. FDR-controlling
False_discovery_rate
Solution of Einstein field equations
closed null curve. At the critical radius r = r c {\displaystyle r=r_{c}} , the vector field ∂ φ {\displaystyle \partial _{\varphi }} becomes null. For
Gödel_metric
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