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Concept in linear algebra
In linear algebra, a reducing subspace W {\displaystyle W} of a linear map T : V → V {\displaystyle T:V\to V} from a Hilbert space V {\displaystyle V}
Reducing_subspace
In mathematics, vector subspace
linear subspace or vector subspace is a vector space that is a subset of some larger vector space. A linear subspace is usually simply called a subspace when
Linear_subspace
decomposition LU decomposition QR decomposition Polar decomposition Reducing subspace Spectral theorem Singular value decomposition Higher-order singular
Outline_of_linear_algebra
Vectors mapped to 0 by a linear map
mapped to the zero vector of the co-domain; the kernel is always a linear subspace of the domain. That is, given a linear map L : V → W between two vector
Kernel_(linear_algebra)
Vector satisfying some of the criteria of an eigenvector
independent generalized eigenvectors which form a basis for an invariant subspace of V. Using generalized eigenvectors, a set of linearly independent eigenvectors
Generalized_eigenvector
estimate an effective dimension reducing subspace, it is now understood that it estimates only the central subspace, which is generally different. More
Sufficient dimension reduction
Sufficient_dimension_reduction
Numerical approximation algorithm
methods are the stationary iterative methods, and the more general Krylov subspace methods. Stationary iterative methods solve a linear system with an operator
Iterative_method
Fundamental space of geometry
subspaces: its Euclidean subspaces and its linear subspaces. Linear subspaces are Euclidean subspaces and a Euclidean subspace is a linear subspace if
Euclidean_space
Method of data analysis
number of possible values with each dimension, complete enumeration of all subspaces becomes intractable with increasing dimensionality. This problem is known
Clustering high-dimensional data
Clustering_high-dimensional_data
Method of data analysis
principal components, and the PCA subspace spanned by the principal directions is identical to the cluster centroid subspace. However, that PCA is a useful
Principal_component_analysis
Algebraic structure in linear algebra
if and only if all its coefficients are zero. Linear subspace A linear subspace or vector subspace W of a vector space V is a non-empty subset of V that
Vector_space
2008 video game
more extensive single-player mode than its predecessors, known as "The Subspace Emissary". This mode is a side-scrolling and plot-driven beat 'em up featuring
Super_Smash_Bros._Brawl
Jargon and esoteric terms used in BDSM
for a specified period (not to be confused with "bottom" or "slave"). Subspace: A psychological state caused by excitement and sense of "letting go" of
Glossary_of_BDSM
Approach to dimensionality reduction
Multilinear subspace learning is an approach for disentangling the causal factor of data formation and performing dimensionality reduction. The Dimensionality
Multilinear_subspace_learning
In linear algebra, generated subspace
linear subspace of V that contains S. It is the set of all finite linear combinations of the elements of S, and the intersection of all linear subspaces that
Linear_span
proves the invariant subspace claim. In fact, one can conclude something stronger. The range of EB is actually a reducing subspace of A, i.e. its orthogonal
Quasinormal_operator
Result about when a matrix can be diagonalized
subspace" consisting of an almost eigenvector, i.e., a closed subspace VE of V associated with a Borel set E⊂σ(A) in the spectrum of A. This subspace
Spectral_theorem
mathematics, the commutator subspace of a two-sided ideal of bounded linear operators on a separable Hilbert space is the linear subspace spanned by commutators
Commutator_subspace
Theorem on extension of bounded linear functionals
allows the extension of bounded linear functionals defined on a vector subspace of some normed vector space to the whole space. In a different guise the
Hahn–Banach_theorem
Completion of the usual space with "points at infinity"
dimension n is defined as the set of the vector lines (that is, vector subspaces of dimension one) in a vector space V of dimension n + 1. Equivalently
Projective_space
Orthonormalization of a set of vectors
generates an orthogonal set S′={𝐮1,…,𝐮k} that spans the same k-dimensional subspace of ℝn as S. The method is named after Jørgen Pedersen Gram and Erhard Schmidt
Gram–Schmidt_process
1997 video game
SubSpace is a 2D space shooter video game created in 1995 and released in 1997 by Virgin Interactive which was a finalist for the Academy of Interactive
SubSpace_(video_game)
Mathematical set with some added structure
structure defining the relationships among the elements of the set. A subspace is a subset of the parent space which retains the same mathematical structure
Space_(mathematics)
Method in machine learning
learning the random subspace method, also called attribute bagging or feature bagging, is an ensemble learning method that attempts to reduce the correlation
Random_subspace_method
Mathematical space
differentiable manifold that parameterizes the set of all k-dimensional linear subspaces of an n-dimensional vector space V over a field K that has a differentiable
Grassmannian
Type of vector space in math
Hilbert space. At a deeper level, perpendicular projection onto a linear subspace plays a significant role in optimization problems and other aspects of
Hilbert_space
of a subnormal A if K' ⊂ K is a reducing subspace of B and H ⊂ K' , then K' = K. (A subspace is a reducing subspace of B if it is invariant under both
Subnormal_operator
the theorem on complete reducibility: the case where a representation V contains a nontrivial, irreducible, invariant subspace W of codimension one. Let
Weyl's theorem on complete reducibility
Weyl's_theorem_on_complete_reducibility
Type of group and algebra representation
G-invariant subspaces, e.g. the whole vector space V, and {0}). If there is a proper nontrivial invariant subspace, ρ is said to be reducible. Group elements
Irreducible_representation
Concept in differential geometry
irreducible as a group representation, or reducible in the sense that there is a splitting of TxM into orthogonal subspaces TxM = T′xM ⊕ T″xM, each of which is
Holonomy
Canadian series of graphic novels
becomes Scott's primary love interest. She is able to use interdimensional "Subspace" to travel long distances quickly often using Scott's head to go through
Scott_Pilgrim
Vector space with generalized dot product
extended by completion to a Hilbert space H‾. This means that H is a linear subspace of H‾, the inner product of H is the restriction of that of H‾, and H is
Inner_product_space
Various meanings of the terms
i.e., non-orthogonal design of modules and interfaces. Orthogonality reduces testing and development time because it is easier to verify designs that
Orthogonality
Iterative method for approximating eigenvectors
non-Hermitian) matrices by constructing an orthonormal basis of the Krylov subspace, which makes it particularly useful when dealing with large sparse matrices
Arnoldi_iteration
Method for solving continuous operator problems (such as differential equations)
unchanged and only the spaces have changed. Reducing the problem to a finite-dimensional vector subspace allows us to numerically compute un as a finite
Galerkin_method
Form of a matrix indicating its eigenvalues and their algebraic multiplicities
This is the invariant subspace decomposition ℂn=⨁iYi given in a previous section. Each ei(T) is the projection onto the subspace spanned by the Jordan
Jordan_normal_form
Bounded operators with sub-unit norm
Γ is completely non-unitary in the sense that it has no non-zero reducing subspaces on which its restriction is unitary. If U = 0, T is said to be a completely
Contraction_(operator_theory)
Transformation of a topological space
point v along the projection p:(X×{0})→v. If X is a non-empty compact subspace of Euclidean space, the cone on X is homeomorphic to the union of segments
Cone_(topology)
Algebraic tool for computing topological spaces' invariants
Mayer and Leopold Vietoris. The method consists of splitting a space into subspaces, for which the homology or cohomology groups may be easier to compute
Mayer–Vietoris_sequence
Square matrix constructed from a monic polynomial
algorithms is given by Frobenius endomorphism Cayley–Hamilton theorem Krylov subspace Horn, Roger A.; Charles R. Johnson (1985). Matrix Analysis. Cambridge,
Companion_matrix
In mathematics, vector space of linear forms
algebraic dual space. When defined for a topological vector space, there is a subspace of the dual space, corresponding to continuous linear functionals, called
Dual_space
Mathematical construct in computer algebra
the geometric operation of projection of an affine algebraic set into a subspace of the ambient space: with above notation, the (Zariski closure of) the
Gröbner_basis
Homology for a pair of topological spaces
mathematics, the (singular) homology of a topological space relative to a subspace is a construction in singular homology, for pairs of spaces. The relative
Relative_homology
Mathematical frame extension
A‖f‖2≤∑i∈ℐvi2‖PWif‖2≤B‖f‖2,∀f∈ℋ, where PWi denotes the orthogonal projection onto the subspace Wi. The constants A and B are called lower and upper bound, respectively
Fusion_frame
Concepts from linear algebra
nullspace is a linear subspace of the domain, E is a linear subspace of ℂn. Because the eigenspace E is a linear subspace, it is closed under addition
Eigenvalues_and_eigenvectors
Geometric transformation that preserves lines but not angles nor the origin
affine space onto itself while preserving both the dimension of any affine subspaces (meaning that it sends points to points, lines to lines, planes to planes
Affine_transformation
Neural network that learns efficient data encoding in an unsupervised manner
vector subspace as the one spanned by the first p principal components, and the output of the autoencoder is an orthogonal projection onto this subspace. The
Autoencoder
Mathematical property
space V is called semi-simple if every T-invariant subspace has a complementary T-invariant subspace. This is equivalent to the minimal polynomial of T
Semi-simplicity
Technique in mathematical modeling
obtain linear approximations in subspaces. Building on nonlinear approximations is essential for efficiently reducing certain problem classes such as
Model_order_reduction
Vector spaces associated to a matrix
The column space of an m×n matrix with components from F is a linear subspace of the m-space Fm. The dimension of the column space is called the rank
Row_and_column_spaces
Algebraic structure designed for geometry
these and several derived operations allow a correspondence of elements, subspaces and operations of the algebra with geometric interpretations. For several
Geometric_algebra
Tree-based ensemble machine learning methods
random decision forests was created in 1995 by Tin Kam Ho using the random subspace method, which, in Ho's formulation, is a way to implement the "stochastic
Random_forest
Number of values in the final calculation of a statistic that are free to vary
of the data vector onto the subspace spanned by the vector of 1's. The 1 degree of freedom is the dimension of this subspace. The second residual vector
Degrees of freedom (statistics)
Degrees_of_freedom_(statistics)
Canonical form of matrices over a field
F. The form reflects a minimal decomposition of the vector space into subspaces that are cyclic for A (i.e., spanned by some vector and its repeated images
Frobenius_normal_form
Locus of the zeros of a polynomial of degree two
(k+1)-dimensional subspace of Vn+1 is a k-dimensional subspace of Pn(K). A 2-dimensional subspace is a plane. In case of n>3 a (n−1)-dimensional subspace is called
Quadric
Measure of the joint variability
vector space is isomorphic to the subspace of random variables with finite second moment and mean zero; on that subspace, the covariance is exactly the L2
Covariance
Branch of mathematics
mathematical structures. These subsets are called linear subspaces. More precisely, a linear subspace of a vector space V over a field F is a subset W of V
Linear_algebra
smoothest invariant manifold serving as the nonlinear extension of a spectral subspace of a linear dynamical system under the addition of nonlinearities. SSM
Spectral_submanifold
Algebra based on a vector space with a quadratic form
unital associative algebra with the additional structure of a distinguished subspace. As K-algebras, they generalize the real numbers, complex numbers, quaternions
Clifford_algebra
Mathematical theory
contractible subspace of the sphere Sn of dimension n. Let Sn∖X be the complement of X in Sn. Then if H~ stands for reduced homology or reduced cohomology
Alexander_duality
Algorithm used for frequency estimation and radio direction finding
\sigma ^{2}} and span the noise subspace U N {\displaystyle {\mathcal {U}}_{N}} , which is orthogonal to the signal subspace, U S ⊥ U N {\displaystyle {\mathcal
MUSIC_(algorithm)
Process in quantum computing
and apply a unitary encoding circuit to rotate the global state into a subspace of a larger Hilbert space. This highly entangled, encoded state corrects
Quantum_error_correction
Matrix representing a Euclidean rotation
space (or subspace). For a 2 × 2 matrix the trace is 2 cos θ, and for a 3 × 3 matrix it is 1 + 2 cos θ. In the three-dimensional case, the subspace consists
Rotation_matrix
Algebraic structure used in analysis
A Lie subalgebra is a linear subspace 𝔥⊆𝔤 which is closed under the Lie bracket. An ideal 𝔦⊆𝔤 is a linear subspace that satisfies the stronger condition
Lie_algebra
Subset (often algebraic set) that is not the union of subsets of the same nature
of two proper closed subsets, and an irreducible component is a maximal subspace (necessarily closed) that is irreducible for the induced topology. Although
Irreducible_component
Grouping a set of objects by similarity
also belong to the parent cluster Subspace clustering: while an overlapping clustering, within a uniquely defined subspace, clusters are not expected to overlap
Cluster_analysis
Area of mathematics
whose coordinates sum to zero, and when n ≥ 2, the representation on this subspace is an (n − 1)-dimensional irreducible representation, called the standard
Representation theory of the symmetric group
Representation_theory_of_the_symmetric_group
The ideal sheaves on a geometric object are closely connected to its subspaces. Let X be a topological space and A a sheaf of rings on X; i.e., let (X
Ideal_sheaf
Dimension of the column space of a matrix
theorem. Alternatively, if M is a linear subspace then dim(AM)≤dim(M); apply this inequality to the subspace defined by the orthogonal complement of
Rank_(linear_algebra)
Group homomorphism into the general linear group over a vector space
permutation of the three variables. For instance, (12) sends x13 to x23. A subspace W of V that is invariant under the group action is called a subrepresentation
Group_representation
Representation of a Lie algebra as a set of linear transformations
is completely reducible if and only if every invariant subspace of V has an invariant complement. (That is, if W is an invariant subspace, then there is
Lie_algebra_representation
Separation process used to purify sea water
heat sink in the other end. Each space consists of two communicating subspaces, the exterior of the tubes of stage n and the interior of the tubes in
Multiple-effect_distillation
Mathematical representation
invariant subspace of H1(Dn) (under the action of Bn) is primitive and infinite cyclic. Let π : H1(Dn) → Z be the projection onto this invariant subspace. Then
Burau_representation
Quotient of two integers
carry an order topology. The rational numbers, as a subspace of the real numbers, also carry a subspace topology. The rational numbers form a metric space
Rational_number
Mathematical space with a notion of distance
topological space, then the subspace consisting of all bounded continuous functions from X to M is also complete. When X is a subspace of ℝn, this function space
Metric_space
Mathematical approach to quantum physics
low-energy subspace ℋL or in the high-energy subspace ℋH, such that there is no matrix element in H(0) connecting the low- and the high-energy subspaces, i.e
Perturbation theory (quantum mechanics)
Perturbation_theory_(quantum_mechanics)
Method for numerical solution of certain systems of equations
equations. The method approximates the solution by the vector in a Krylov subspace with minimal residual. The Arnoldi iteration is used to find this vector
Generalized minimal residual method
Generalized_minimal_residual_method
Function over linear operators
trace of ρ with respect to the system B, denoted by ρA, is called the reduced state of ρ on system A. In symbols,ρA=TrBρ. To show that this is indeed
Partial_trace
American mathematician
297–301. Cooley, James W., Timothy M. Toolan and Donald W. Tufts. "A Subspace Tracking Algorithm Using the Fast Fourier Transform." IEEE Signal Processing
James_Cooley
Branch of topology
that generates them. Every subset of a topological space can be given the subspace topology in which the open sets are the intersections of the open sets
General_topology
1999 space combat simulation computer game
jump nodes and traveling through subspace, while shorter intra-system distances are done by "hopping" into subspace at any time. All ships in a mission
FreeSpace_2
handle. Subspace weapons are a class of directed energy weapons that directly affect subspace. The weapons can produce actual tears in subspace, and are
Weapons_in_Star_Trek
Quantum search algorithm
that the quantum state of Grover's algorithm stays in a two-dimensional subspace after each step. Consider the plane spanned by |s⟩ and |ω⟩; equivalently
Grover's_algorithm
Robot used in manufacturing
robot and the 3 orientation coordinates are in the constraint subspace. The motion subspace of lower mobility manipulators may be further decomposed into
Industrial_robot
Embedding of a Grassmannian into projective space
map embeds the Grassmannian Gr(k,V), whose elements are k-dimensional subspaces of an n-dimensional vector space V, either real or complex, in a projective
Plücker_embedding
Theorem used in quantum mechanics for angular momentum calculations
is a theorem that tells how vector operators behave in a subspace. Within a given subspace, a component of a vector operator will behave in a way proportional
Wigner–Eckart_theorem
Most widely known generalized inverse of a matrix
annihilates the kernel of A and acts as a traditional inverse of A on the subspace orthogonal to the kernel. In the following discussion, the following conventions
Moore–Penrose_inverse
Differential geometry construct on fiber bundles
For each point e∈E, He is a vector subspace of the tangent space TeE to E at e, called the horizontal subspace of the connection at e. He depends smoothly
Ehresmann_connection
Branch of algebraic geometry
sets in a Grassmannian defined by conditions of incidence of a linear subspace in projective space with a given flag. For further details see Schubert
Schubert_calculus
Python library for machine learning
the library for both recommendation systems (e.g., for Digg) and dynamic subspace clustering applied to weather forecasting data. PeerIndex used scikit-learn
Scikit-learn
Method of solving differential equations
the subspace correction framework, BPX preconditioner is a parallel subspace correction method whereas the classic V-cycle is a successive subspace correction
Multigrid_method
Process of reducing the number of random variables under consideration
representation can be used in dimensionality reduction through multilinear subspace learning. The main linear technique for dimensionality reduction, principal
Dimensionality_reduction
Topics referred to by the same term
file shares Cosworth DFS, a motor racing engine Decoherence-free subspaces, subspace of a system's Hilbert space where the system is decoupled from the
DFS
Matrix decomposition
k times has a k-dimensional subspace of corresponding left-singular vectors. Any orthonormal basis for this subspace can be taken as the corresponding
Singular_value_decomposition
Method for finding largest (or smallest) eigenvalues
problems was described in. Local minimization of the Rayleigh quotient on the subspace spanned by the current approximation, the current residual and the previous
LOBPCG
Mathematical operation on vector spaces
nonzero at an only a finite number of elements of BV×BW, and consider the subspace of such maps instead. In either construction, the tensor product of two
Tensor_product
Vectors whose linear combinations are nonzero
vector subspaces M and N of a vector space X are said to be linearly independent if M∩N={0}. More generally, a collection M1,…,Md of subspaces of X are
Linear_independence
Real square matrix whose columns and rows are orthogonal unit vectors
completely characterized by one angle, and may affect more than one planar subspace. It is common to describe a 3 × 3 rotation matrix in terms of an axis and
Orthogonal_matrix
runabout, which is shrunken to four inches long as they investigate a rare subspace compression phenomenon. 139 15 "Honor Among Thieves" Unknown Allan Eastman
List of Star Trek: Deep Space Nine episodes
List_of_Star_Trek:_Deep_Space_Nine_episodes
Concept in machine learning
(multilinear) factor analysis disentangles and reduces the influence of different causal factors with multilinear subspace learning. When treating an image or a
Tensor_(machine_learning)
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