mathematical fields of linear algebra and functional analysis, the orthogonal complement of a subspace W {\displaystyle W} of a vector space V {\displaystyle...

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...

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...

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...

every vector to an orthogonal vector but is not identically 0 {\displaystyle 0} . Orthogonal complement The orthogonal complement of a subset C ⊆ V {\displaystyle...

(sometimes called an antonym) Complement (group theory) Complementary subspaces Orthogonal complement Schur complement Complement (complexity), relating to...

a right angle, whereas orthogonal is used in generalizations, such as orthogonal vectors or orthogonal curves. Orthogonality is also used with various...

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...

the orthogonal complement operation, provides an example of an orthocomplemented lattice that is not, in general, distributive. Some complemented lattices...

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...

vector spaces, for example, orthogonal complements exist. However, these spaces may have null vectors that are orthogonal to themselves, and consequently...

{\displaystyle P} ⁠ is the orthogonal projector onto the range of ⁠ A {\displaystyle A} ⁠ (which equals the orthogonal complement of the kernel of ⁠ A ∗ {\displaystyle...

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...

corresponding to different eigenvalues are orthogonal, and a normal operator stabilizes the orthogonal complement of each of its eigenspaces. This implies...

_{2}+M_{X_{1}}u,} where M X 1 {\displaystyle M_{X_{1}}} projects onto the orthogonal complement of the image of the projection matrix X 1 ( X 1 T X 1 ) − 1 X 1...

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...

  Orthogonal complement: If W is a linear subspace of an inner product space V, then W ⊥ {\displaystyle W^{\bot }} denotes its orthogonal complement, that...

collinear. The intersections of any Euclidean linear subspace with its orthogonal complement is the {0} subspace. But the definition from the previous subsection...

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...

In mathematics, the orthogonal group in dimension n, denoted O(n), is the group of distance-preserving transformations of a Euclidean space of dimension...

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...

furthermore, the orthogonal complement L⊥ of L is also invariant under T. For example, the space H can be decomposed as the orthogonal direct sum of two...

algebra and acts absolutely irreducibly on the 196883-dimensional orthogonal complement of this 1-space. (The Monster preserves the standard inner product...

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...

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...

In mathematics, an orthogonal array (more specifically, a fixed-level orthogonal array) is a "table" (array) whose entries come from a fixed finite set...

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...

to some eigenspace Vλ. Let Vλ⊥ be its orthogonal complement. It is clear that, with respect to this orthogonal decomposition, A has matrix representation...

the isotropy group is parametrized by the unitary matrices on the orthogonal complement of | ψ ⟩ {\displaystyle |\psi \rangle } , which is isomorphic to...

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 ) )...