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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)
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
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
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)
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
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
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
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
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
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
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
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)
Concept in mathematics
nontrivial. If W is a subset of V, then its orthogonal complement W⊥ is the set of all vectors in V that are orthogonal to every vector in W; it is a subspace
Symmetric_bilinear_form
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)
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
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
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
(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
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
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
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
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
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
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
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
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
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
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
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
Euclidean space Orthogonality Orthogonal complement Orthogonal projection Orthogonal group Pseudo-Euclidean space Null vector Indefinite orthogonal group Orientation
Outline_of_linear_algebra
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 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
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
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)
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
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
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)
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
Concept in geometry
}\cap G} , where superscript ⊥ {\displaystyle \bot } denotes the orthogonal complement. The variational characterization of singular values and vectors
Angles_between_flats
}} 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
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
Representation of a quantum mechanical system
the isotropy group is parametrized by the unitary matrices on the orthogonal complement of | ψ ⟩ {\displaystyle |\psi \rangle } , which is isomorphic to
Bloch_sphere
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
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
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
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
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
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)
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
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)
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
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
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
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
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
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
Sporadic simple group
acts on a 23-dimensional even lattice with no roots, given by the orthogonal complement of a norm 6 vector of the Leech lattice. This gives 23-dimensional
Conway_group_Co3
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
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
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
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
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
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
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
Mathematical concept
transformation that fixes p and is the negative identity on the orthogonal complement of the line represented by p. Through any two points p, q in complex
Complex_projective_space
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
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
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
\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
Statistical learning theory
\ldots ,\varphi (x_{n})\right\rbrace } , and the other lying in the orthogonal complement: f = ∑ i = 1 n α i φ ( x i ) + v , {\displaystyle f=\sum _{i=1}^{n}\alpha
Representer_theorem
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
{\displaystyle V} be the orthogonal complement of H 1 {\displaystyle {\mathcal {H}}_{1}} in H {\displaystyle {\mathcal {H}}} . Since orthogonality implies independence
Gaussian_probability_space
{\displaystyle \{v_{1}\}^{\perp }} , where ⊥ {\displaystyle \perp } denotes orthogonal complement. By Hermiticity of T, this is the same as Ran(T) ⊂ {\displaystyle
Range_criterion
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
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
Concept in functional analysis
a TVS are closed, but those that are, do have complements. In a Hilbert space, the orthogonal complement M ⊥ {\displaystyle M^{\bot }} of any closed vector
Complemented_subspace
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
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
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)
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
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
Result on the topology of operators on an infinite-dimensional, complex Hilbert space
single vector v of the unit sphere is the unitary group of the orthogonal complement of v; therefore the homotopy long exact sequence predicts that all
Kuiper's_theorem
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
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
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
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
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
(a Borel subalgebra) of g {\displaystyle {\mathfrak {g}}} ; the orthogonal complement with respect to the Killing form of p {\displaystyle {\mathfrak
Parabolic_Lie_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
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)
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
Representations of finite groups, particularly on vector spaces
Restricted to this subspace we obtain the trivial representation. The orthogonal complement of C ( e 1 + e 2 + e 3 ) {\displaystyle \mathbb {C} (e_{1}+e_{2}+e_{3})}
Representation theory of finite groups
Representation_theory_of_finite_groups
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
Linear operator defined on a dense linear subspace
: H 1 → H 2 {\displaystyle T:H_{1}\to H_{2}} coincides with the orthogonal complement of the range of the adjoint. That is, ker ( T ) = ran ( T ∗
Unbounded_operator
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)
Type of smooth complex surface of kodaira dimension 0
connected component of the complement of these hyperplanes in the positive cone. Any two such components are isomorphic via the orthogonal group of the lattice
K3_surface
entries are the squares of the absolute values of the entries of some orthogonal matrix Precision matrix — a symmetric n×n matrix, formed by inverting
List_of_named_matrices
ORTHOGONAL COMPLEMENT
ORTHOGONAL COMPLEMENT
ORTHOGONAL COMPLEMENT
ORTHOGONAL COMPLEMENT
Boy/Male
Muslim/Islamic
Wisdom Learning, Science
Girl/Female
Indian
Evening time
Boy/Male
Scandinavian
Wolf counsel.
Boy/Male
Muslim
One who confirms, Verifies another (1)
Boy/Male
Arabic, Muslim
The Awaiting
Boy/Male
English
keeper of the keys; pure.' Also Kayla is a character on daytime TV 'Days of Our Lives.
Boy/Male
Tamil
Love to God
Girl/Female
Indian
Ready for battle
Boy/Male
American, Australian, British, Celtic, Christian, English, Indian, Irish
Little Fire; Born of Fire
Girl/Female
Christian & English(British/American/Australian)
Pretty
ORTHOGONAL COMPLEMENT
ORTHOGONAL COMPLEMENT
ORTHOGONAL COMPLEMENT
ORTHOGONAL COMPLEMENT
ORTHOGONAL COMPLEMENT
a.
Serving to fill out or to complete; as, complementary numbers.
v. t.
The interval wanting to complete the octave; -- the fourth is the complement of the fifth, the sixth of the third.
v. t.
To unite in couples; to form a pair of; to bring together, as things which belong together, or which complement, or are adapted to one another.
n.
A system of jurisprudence, supplemental to law, properly so called, and complemental of it.
n.
The space included between the boundary lines of two similar parallelograms, the one within the other, with an angle in common; as, the gnomon bcdefg of the parallelograms ac and af. The parallelogram bf is the complement of the parallelogram df.
v. t.
To match; to mate in contest; to furnish a complement to; as, to cap text; to cap proverbs.
n.
A rectangular figure.
a.
Not having the full complement of tones; -- said of a chord of only two tones, which requires a third tone to be sounded with them to make the combination pleasing to the ear; as, a naked fourth or fifth.
n.
The instruments employed by a full band, collectively; as, an orchestra of forty stringed instruments, with proper complement of wind instruments.
v. t.
To supply with men; to furnish with a sufficient force or complement of men, as for management, service, defense, or the like; to guard; as, to man a ship, boat, or fort.
n.
A luminous appearance, or an image seen after the eye has been exposed to an intense light or a strongly illuminated object. When the object is colored, the image appears of the complementary color, as a green image seen after viewing a red wafer lying on white paper. Called also ocular spectrum.
a.
Right-angled; rectangular; as, an orthogonal intersection of one curve with another.
a.
Not having a full complement of men; as, a vessel light-handed.
a.
See Octagonal.
a.
Pertaining to, or evincing, orthodoxy; orthodox.
adv.
Perpendicularly; at right angles; as, a curve cuts a set of curves orthogonally.
v. t.
To assume; to adopt; to acquire, as shape; to permit to one's self; to indulge or engage in; to yield to; to have or feel; to enjoy or experience, as rest, revenge, delight, shame; to form and adopt, as a resolution; -- used in general senses, limited by a following complement, in many idiomatic phrases; as, to take a resolution; I take the liberty to say.
n.
A band composed, for the largest part, of players of the various viol instruments, many of each kind, together with a proper complement of wind instruments of wood and brass; -- as distinguished from a military or street band of players on wind instruments, and from an assemblage of solo players for the rendering of concerted pieces, such as septets, octets, and the like.
n.
A number of things of the same kind, ordinarily used or classed together; a collection of articles which naturally complement each other, and usually go together; an assortment; a suit; as, a set of chairs, of china, of surgical or mathematical instruments, of books, etc.