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Concept in linear algebra
mathematical fields of linear algebra and functional analysis, the orthogonal complement of a subspace W {\displaystyle W} of a vector space V {\displaystyle
Orthogonal_complement
Idempotent linear transformation from a vector space to itself
range (which is a complement of the kernel). When these basis vectors are orthogonal to the kernel, then the projection is an orthogonal projection. When
Projection_(linear_algebra)
Generalization of perpendicularity
largest subspace of V {\displaystyle V} that is orthogonal to a given subspace is its orthogonal complement. Given a module M {\displaystyle M} and its dual
Orthogonality_(mathematics)
Type of vector space in math
characterized in terms of the orthogonal complement: if V is a subspace of H, then the closure of V is equal to V⊥⊥. The orthogonal complement is thus a Galois connection
Hilbert_space
Various meanings of the terms
Orthogonality is a term with various meanings depending on the context. In mathematics, orthogonality is the generalization of the geometric notion of
Orthogonality
Vector space with generalized dot product
every vector to an orthogonal vector but is not identically 0 {\displaystyle 0} . Orthogonal complement The orthogonal complement of a subset C ⊆ V {\displaystyle
Inner_product_space
also used. □⊥ 1. Orthogonal complement: If W is a linear subspace of an inner product space V, then W⊥ denotes its orthogonal complement, that is, the linear
Glossary of mathematical symbols
Glossary_of_mathematical_symbols
Mathematical space
V} into the orthogonal direct sum V = w ⊕ w ⊥ {\displaystyle V=w\oplus w^{\perp }} of w {\displaystyle w} and its orthogonal complement w ⊥ {\displaystyle
Grassmannian
In mathematics, vector subspace
vector spaces, for example, orthogonal complements exist. However, these spaces may have null vectors that are orthogonal to themselves, and consequently
Linear_subspace
Most widely known generalized inverse of a matrix
{\displaystyle P} is the orthogonal projector onto the range of A {\displaystyle A} (which equals the orthogonal complement of the kernel of A ∗ {\displaystyle
Moore–Penrose_inverse
Hilbert spaces such that it is an isometry on the orthogonal complement of its kernel. The orthogonal complement of its kernel is called the initial subspace
Partial_isometry
Space in mathematics and theoretical physics
collinear. The intersections of any Euclidean linear subspace with its orthogonal complement is the {0} subspace. But the definition from the previous subsection
Pseudo-Euclidean_space
Bound lattice in which every element has a complement
the orthogonal complement operation, provides an example of an orthocomplemented lattice that is not, in general, distributive. Some complemented lattices
Complemented_lattice
Vectors mapped to 0 by a linear map
ker ( L ) {\displaystyle V/\ker(L)} can be identified with the orthogonal complement in V of ker ( L ) {\displaystyle \ker(L)} . This is the generalization
Kernel_(linear_algebra)
(on a complex Hilbert space) continuous linear operator
corresponding to different eigenvalues are orthogonal, and a normal operator stabilizes the orthogonal complement of each of its eigenspaces. This implies
Normal_operator
Topics referred to by the same term
(sometimes called an antonym) Complement (group theory) Complementary subspaces Orthogonal complement Schur complement Complement (complexity), relating to
Complement
Type of group in mathematics
In mathematics, the orthogonal group in dimension n, denoted O(n), is the group of distance-preserving transformations of a Euclidean space of dimension
Orthogonal_group
Mathematical operation on vectors in 3D space
The cross product a × b is defined as a vector c that is perpendicular (orthogonal) to both a and b, with a direction given by the right-hand rule and a
Cross_product
Vector spaces associated to a matrix
only if x is orthogonal (perpendicular) to each of the row vectors of A. It follows that the null space of A is the orthogonal complement to the row space
Row_and_column_spaces
Theorem in statistics and econometrics
case) the three step process: Project X {\displaystyle X} onto the orthogonal complement of Col ( Z ) {\displaystyle {\text{Col}}(Z)} , obtaining residuals
Frisch–Waugh–Lovell_theorem
Mapping from a Euclidean space to itself
be described either by the subspace that remains fixed or by its orthogonal complement, whose vectors are reversed. In the preceding two-dimensional section
Reflection_(mathematics)
Mathematical inequality relating inner products and norms
an eigenvector, then the spectral theorem follows by taking the orthogonal complement and arguing by induction on the dimension of the inner product space
Cauchy–Schwarz_inequality
Concept in mathematics
V} , then its orthogonal complement W ⊥ {\displaystyle W^{\perp }} is the set of all vectors in V {\displaystyle V} that are orthogonal to every vector
Symmetric_bilinear_form
theorem in linear algebra is as follows: if M is a matrix, then the orthogonal complement of the row space of M is the null space of M: ( row M ) ⊥ = ker
Fredholm's_theorem
Particular correspondence between two partially ordered sets
fundamental group. Given an inner product space V, we can form the orthogonal complement F(X ) of any subspace X of V. This yields an antitone Galois connection
Galois_connection
Conjugate transpose of an operator in infinite dimensions
y=A^{*}x\}\subseteq H\oplus H} of A ∗ {\displaystyle A^{*}} is the orthogonal complement of J G ( A ) : {\displaystyle JG(A):} G ( A ∗ ) = ( J G ( A ) )
Hermitian_adjoint
Type of mathematical array
In mathematics, an orthogonal array (more specifically, a fixed-level orthogonal array) is a table ("array") whose entries come from a fixed finite set
Orthogonal_array
Type of high-dimensional algebra
group. The 196883-dimensional subspace ( W {\displaystyle W} ): The orthogonal complement, where the Monster acts absolutely irreducibly. This dimension relates
Griess_algebra
Linear subspace generated from a vector acted on by a power series of a matrix
so the uncontrollable and unobservable subspaces are simply the orthogonal complement to the Krylov subspace. Modern iterative methods such as Arnoldi
Krylov_subspace
Theorem about the dual of a Hilbert space
\|y\|\leq \|y+sx\|} for all scalars s . {\displaystyle s.} The orthogonal complement of a subset X ⊆ H {\displaystyle X\subseteq H} is X ⊥ := { y ∈ H
Riesz_representation_theorem
All even-degree subgraphs of a graph
(the family of edge sets that span the cuts of the graph) are the orthogonal complements of each other within the edge space. This means that a set S {\displaystyle
Cycle_space
Dimension of the column space of a matrix
\dim(M)} ; apply this inequality to the subspace defined by the orthogonal complement of the image of B C {\displaystyle BC} in the image of B {\displaystyle
Rank_(linear_algebra)
Operation in abstract algebra
reconstruction of a finite vector space from any subspace W and its orthogonal complement: R n = W ⊕ W ⊥ {\displaystyle \mathbb {R} ^{n}=W\oplus W^{\perp
Direct_sum_of_modules
Location of a discrete degeneracy between two electronic states
The space of non-degeneracy-lifting displacements, which is the orthogonal complement of the branching space, is termed the seam space. Movement within
Conical_intersection
Manifold
natural almost complex structure arising from the fact that it is the orthogonal complement of i in the unit sphere of the octonions, but this is not a complex
Complex_manifold
Type of matrix representation
extended by continuity to the closure of Ran(B), and by zero on the orthogonal complement to all of H. The lemma then follows since A*A ≤ B*B implies ker(B)
Polar_decomposition
Matrix factorisation in mathematics
to some eigenspace Vλ. Let Vλ⊥ be its orthogonal complement. It is clear that, with respect to this orthogonal decomposition, A has matrix representation
Schur_decomposition
Concept in geometry
}\cap G} , where superscript ⊥ {\displaystyle \bot } denotes the orthogonal complement. The variational characterization of singular values and vectors
Angles_between_flats
Functional analysis concept
of which corresponding to a real eigenvalue. More precisely, the orthogonal complement of the kernel of T {\displaystyle T} admits an orthonormal basis
Compact operator on Hilbert space
Compact_operator_on_Hilbert_space
Concept in linear algebra
to itself is an invariant subspace of T {\displaystyle T} whose orthogonal complement W ⊥ {\displaystyle W^{\perp }} is also an invariant subspace of
Reducing_subspace
Euclidean space Orthogonality Orthogonal complement Orthogonal projection Orthogonal group Pseudo-Euclidean space Null vector Indefinite orthogonal group Orientation
Outline_of_linear_algebra
Result about when a matrix can be diagonalized
{\displaystyle {\mathcal {K}}^{n-1}={\text{span}}(v_{1})^{\perp }} , the orthogonal complement of v1. By Hermiticity, K n − 1 {\displaystyle {\mathcal {K}}^{n-1}}
Spectral_theorem
Manifold with inversion symmetry
{\displaystyle {\mathfrak {m}}_{a}} are the subspaces fixed by Ad a. Let the orthogonal complement of k a {\displaystyle {\mathfrak {k}}_{a}} in k {\displaystyle {\mathfrak
Hermitian_symmetric_space
Representation of a quantum mechanical system
multiply it by a phase, and it acts as an arbitrary unitary on the orthogonal complement. The pure state space can therefore be identified with the compact
Bloch_sphere
Mathematical concept
}&=W\\\dim W+\dim W^{\perp }&=\dim V.\end{aligned}}} However, unlike orthogonal complements, W⊥ ∩ W need not be 0. We distinguish four cases: W is symplectic
Symplectic_vector_space
Design method of discrete wavelet transforms
(closed) linear hull of the mother wavelet's integer shifts, is the orthogonal complement to V 0 {\displaystyle V_{0}} inside V − 1 {\displaystyle V_{-1}}
Multiresolution_analysis
Vector space consisting of affine subsets
Hilbert space, then the quotient space X/M is isomorphic to the orthogonal complement of M. The quotient of a locally convex space by a closed subspace
Quotient space (linear algebra)
Quotient_space_(linear_algebra)
Area of mathematics
subrepresentation consisting of vectors whose coordinates are all equal. The orthogonal complement consists of those vectors whose coordinates sum to zero, and when
Representation theory of the symmetric group
Representation_theory_of_the_symmetric_group
Function for integral Fourier-like transform
W_{1},W_{0},W_{-1},\dots } are the orthogonal "differences" of the above sequence, that is, Wm is the orthogonal complement of Vm inside the subspace Vm−1
Wavelet
Matrix decomposition
{\displaystyle \mathbf {M} } . The same calculation performed on the orthogonal complement of u {\displaystyle \mathbf {u} } gives the next largest eigenvalue
Singular_value_decomposition
Property of measure-preserving dynamical systems
Let L 0 2 ( X , μ ) {\displaystyle L_{0}^{2}(X,\mu )} denote the orthogonal complement of the constant functions, L 0 2 ( X , μ ) = { f ∈ L 2 ( X , μ )
Ergodicity
Mathematical approach to quantum physics
where the | k ( 0 ) ⟩ {\displaystyle |k^{(0)}\rangle } are in the orthogonal complement of | n ( 0 ) ⟩ {\displaystyle |n^{(0)}\rangle } , i.e., the other
Perturbation theory (quantum mechanics)
Perturbation_theory_(quantum_mechanics)
Dual pair of vector spaces
S} . The definition of a subset being orthogonal to a vector is defined analogously. The orthogonal complement or annihilator of a subset R ⊆ X {\displaystyle
Dual_system
(a Borel subalgebra) of g {\displaystyle {\mathfrak {g}}} ; the orthogonal complement with respect to the Killing form of p {\displaystyle {\mathfrak
Parabolic_Lie_algebra
Differential geometry topic
{\displaystyle {\tilde {G}}_{k,n}\cong {\tilde {G}}_{n-k,n}} via orthogonal complement. In Euclidean 3-space, this says that an oriented 2-plane is characterized
Gauss_map
Partition of a graph's nodes into 2 disjoint subsets
difference of two cut sets as the vector addition operation, and is the orthogonal complement of the cycle space. If the edges of the graph are given positive
Cut_(graph_theory)
Scalar-valued bilinear function
bilinear form is nondegenerate. Suppose W is a subspace. Define the orthogonal complement W ⊥ = { v ∣ B ( v , w ) = 0 for all w ∈ W } . {\displaystyle W^{\perp
Bilinear_form
Equivalence of matrices under row operations
reduced row echelon form. Because the null space of a matrix is the orthogonal complement of the row space, two matrices are row equivalent if and only if
Row_equivalence
Theory of logic to account for observations from quantum theory
of closed subspaces of H; the negation of a proposition V is the orthogonal complement V⊥. The space Q of quantum propositions is also sequentially complete:
Quantum_logic
Type of geometric transformation
0\\0&0&0&0&1\end{pmatrix}}.} This matrix shears parallel to the orthogonal complement of the fourth dimension and in the direction of the x axis of the
Shear_mapping
Mathematical study of linear operators
extended by continuity to the closure of Ran(B), and by zero on the orthogonal complement of Ran(B). The operator C is well-defined since A*A ≤ B*B implies
Operator_theory
\scriptstyle F} , i.e. the determinant of the derivative restricted to the orthogonal complement of its kernel. Note that from Sard's lemma almost every point y
Smooth_coarea_formula
Generalization of complex inner products
sesquilinear form φ over a module M and a subspace (submodule) W of M, the orthogonal complement of W with respect to φ is W ⊥ = { v ∈ M ∣ φ ( v , w ) = 0 , ∀
Sesquilinear_form
Types of mappings in mathematics
{\displaystyle X,} called the null space or kernel of the functional, or the orthogonal complement of x → , {\displaystyle {\vec {x}},} denoted { x → } ⊥ . {\displaystyle
Functional_(mathematics)
Path of an object through spacetime
simultaneity is a statement that N depends on v. Indeed, N is the orthogonal complement of v with respect to η. When two world lines u and w are related
World_line
Non-tensorial representation of the spin group
a maximal isotropic subspace with W ∩ W′ = 0, and let U be the orthogonal complement of W ⊕ W′. In both the even- and odd-dimensional cases W and W′
Spinor
Ideal that maps to zero a subset of a module
the map V × V → K {\displaystyle V\times V\to K} is called the orthogonal complement. Given a module M over a Noetherian commutative ring R, a prime
Annihilator_(ring_theory)
Integral transform and linear operator
) {\displaystyle \operatorname {H} ^{2}(\mathbb {R} )} and its orthogonal complement are eigenspaces of H for the eigenvalues ±i. In other words, H commutes
Hilbert_transform
Concerns the decomposition of representations of a finite group into irreducible pieces
{\displaystyle \mathbb {C} } by constructing U {\displaystyle U} as the orthogonal complement of W {\displaystyle W} under this inner product. One of the approaches
Maschke's_theorem
Algebraic surface defined by a cubic polynomial
connection is that the E 6 {\displaystyle E_{6}} lattice arises as the orthogonal complement to the anticanonical class − K X {\displaystyle -K_{X}} in the Picard
Cubic_surface
Symmetric bilinear form in mathematics
zero intersection, then I and J are orthogonal subspaces with respect to the Killing form. The orthogonal complement with respect to B of an ideal is again
Killing_form
Manifold of all orthonormal k-frames in n-dimensional Euclidean space
the subgroup isomorphic to O(n−k) which acts nontrivially on the orthogonal complement of the space spanned by that frame. Likewise the unitary group U(n)
Stiefel_manifold
Cohomology with real coefficients computed using differential forms
are orthogonal; the orthogonal complement then consists of forms that are both closed and co-closed: that is, of harmonic forms. Here, orthogonality is
De_Rham_cohomology
Algebraic structure in mathematics
algebra generated by V* and with quadratic relations forming the orthogonal complement of S in V* ⊗ V*. A quadratic algebra may be a filtered algebra generated
Quadratic_algebra
with Cn ⊕ (0) and (0) ⊕ Cn and V0 is their orthogonal complement in V. Similarly the orthogonal complement U of V can be written U = U0 ⊕ U1 ⊕ U2. Thus
Mutation_(Jordan_algebra)
Branch of algebraic topology
bundle can be completed to a trivial bundle by summing with its orthogonal complement. Alternatively, K ~ ( X ) {\displaystyle {\widetilde {K}}(X)} can
Topological_K-theory
Sum of directed areas in exterior algebra
Hodge dual relates the blade that represents a subspace to its orthogonal complement, so if a bivector represents a plane then the axial vector associated
Bivector
Quadratic form for which there is a non-zero vector on which the form evaluates to zero
(or metabolic) if there is a subspace which is equal to its own orthogonal complement; equivalently, the index of isotropy is equal to half the dimension
Isotropic_quadratic_form
In mathematics, vector space of linear forms
dimensional vector spaces, the annihilator is dual to (isomorphic to) the orthogonal complement. The annihilator of a subset is itself a vector space. The annihilator
Dual_space
Vector satisfying some of the criteria of an eigenvector
Euclidean space Orthogonality Orthogonal complement Orthogonal projection Orthogonal group Pseudo-Euclidean space Null vector Indefinite orthogonal group Orientation
Generalized_eigenvector
Branch of mathematics that studies abstract algebraic structures
semisimple, since Maschke's result can be proven by taking the orthogonal complement of a subrepresentation. When studying representations of groups
Representation_theory
Concept in mathematics
reducible, in the sense that for any closed invariant subspace, the orthogonal complement is again a closed invariant subspace. This is at the level of an
Unitary_representation
determinant 2. In this correspondence, the lattice L is isomorphic to the orthogonal complement of the vector v. There are 665 orbits of vectors v of norm –4, corresponding
II25,1
Measure used in functional analysis
(E_{2})} are orthogonal to each other. Let V E = im ( π ( E ) ) {\displaystyle V_{E}=\operatorname {im} (\pi (E))} and its orthogonal complement V E ⊥ =
Projection-valued_measure
Complex-valued smooth functions of the upper half plane (harmonic analysis topic)
k)} and has a discrete spectrum. The spectrum belonging to the orthogonal complement has a continuous part and can be described with the help of (modified)
Maass_wave_form
Non-commutative group with 6 elements
and ρ 2 {\displaystyle \rho _{2}} is the representation on its orthogonal complement, which are vectors of the form ( λ 1 , λ 2 , − λ 1 − λ 2 ) {\displaystyle
Dihedral_group_of_order_6
Concept that permeates much of inferential statistics and descriptive statistics
Euclidean space Expected mean squares Orthogonality Orthonormal basis Orthogonal complement, the closed subspace orthogonal to a set (especially a subspace)
Partition_of_sums_of_squares
Concept in mathematics
For a Riemannian manifold one can identify this quotient with the orthogonal complement, but in general one cannot (such a choice is equivalent to a section
Normal_bundle
Wavelet constructed using a spline function
(this closure is denoted by W 0 {\displaystyle W_{0}} ) is the orthogonal complement of V 0 {\displaystyle V_{0}} in V 1 {\displaystyle V_{1}} . The
Spline_wavelet
Mathematical concept
{\displaystyle \mathbb {R} } 7 with the imaginary octonions (the orthogonal complement of the real part of O {\displaystyle \mathbb {O} } ), the cross
Seven-dimensional cross product
Seven-dimensional_cross_product
Mathematical theorem
}}v=0.} For any subspace S let S ⊥ {\displaystyle S^{\bot }} be the orthogonal complement of S. Call the subspace "closed" if S ⊥ ⊥ = S . {\displaystyle S^{\bot
Solèr's_theorem
every ideal J. An ideal I is essential if and only if I⊥, the "orthogonal complement" of I in the Hilbert C*-module B is {0}. Let A be a C*-algebra.
Multiplier_algebra
Particular projective representations of the orthogonal or special orthogonal groups
representations of the orthogonal or special orthogonal groups in arbitrary dimension and signature (i.e., including indefinite orthogonal groups). More precisely
Spin_representation
vector Orthonormal basis Orthogonal complement Orthogonalization Parallelogram law Normal matrix, normal operator Orthogonal matrix Unitary matrix Semi-Hilbert
List of functional analysis topics
List_of_functional_analysis_topics
Non-associative algebras with positive-definite quadratic form
Clifford algebras. Indeed, taking an orthonormal basis ei of the orthogonal complement of 1 gives rise to operators Ui = L(ei) satisfying U i 2 = − I
Hurwitz's theorem (composition algebras)
Hurwitz's_theorem_(composition_algebras)
Statistical property
{\displaystyle {\vec {u}}=(1,\ldots ,1)} and to that direction's orthogonal complement hyperplane. One gets A → = ( X ¯ − μ , … , X ¯ − μ ) {\displaystyle
Bias_of_an_estimator
Theorem in functional analysis
Analogously, consider now a (k − 1)-dimensional subspace Sk−1, whose the orthogonal complement is denoted by Sk−1⊥. If S' = span{u1...uk}, S ′ ∩ S k − 1 ⊥ ≠ 0
Min-max_theorem
Constructions in nonsmooth analysis
of a normal cone to a subset of a space generalizes that of the orthogonal complement / annihilator of a vector space, (outward) normal vector fields
Normal cone (variational analysis)
Normal_cone_(variational_analysis)
}} denotes the subbundle of T M {\displaystyle TM} that is the orthogonal complement of k e r ( d f x ) ⊂ T x M {\displaystyle \mathrm {ker} (df_{x})\subset
Riemannian_submersion
Number of values in the final calculation of a statistic that are free to vary
vector is the least-squares projection onto the (n − 1)-dimensional orthogonal complement of this subspace, and has n − 1 degrees of freedom. In statistical
Degrees of freedom (statistics)
Degrees_of_freedom_(statistics)
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ORTHOGONAL COMPLEMENT
ORTHOGONAL COMPLEMENT
ORTHOGONAL COMPLEMENT
ORTHOGONAL COMPLEMENT
ORTHOGONAL COMPLEMENT
ORTHOGONAL COMPLEMENT
ORTHOGONAL COMPLEMENT
ORTHOGONAL COMPLEMENT
ORTHOGONAL COMPLEMENT
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