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Vector satisfying some of the criteria of an eigenvector
In linear algebra, a generalized eigenvector of an n × n {\displaystyle n\times n} matrix A {\displaystyle A} is a vector which satisfies certain criteria
Generalized_eigenvector
Matrix decomposition
{\displaystyle k} . That is, it is the space of generalized eigenvectors (in the first sense), where a generalized eigenvector is any vector which eventually becomes
Eigendecomposition of a matrix
Eigendecomposition_of_a_matrix
Numerical methods for matrix eigenvalue calculation
also find eigenvectors. Given an n × n square matrix A of real or complex numbers, an eigenvalue λ and its associated generalized eigenvector v are a pair
Eigenvalue_algorithm
Mathematical optimization concept
solution to a linear programming problem can be regarded as a generalized eigenvector. The eigenequations of a square matrix are as follows: p T A =
Dual_linear_program
Concepts from linear algebra
matrices, the notion of eigenvectors generalizes to generalized eigenvectors and the diagonal matrix of eigenvalues generalizes to the Jordan normal form
Eigenvalues_and_eigenvectors
Form of a matrix indicating its eigenvalues and their algebraic multiplicities
the complex Jordan form. For a real matrix the nonreal eigenvectors and generalized eigenvectors can always be chosen to form complex conjugate pairs.
Jordan_normal_form
Basis of a type of algebraic structure
In linear algebra, it refers to a set of n linearly independent generalized eigenvectors of an n×n matrix A {\displaystyle A} , if the set is composed entirely
Canonical_basis
Result about when a matrix can be diagonalized
delta-functions are "generalized eigenvectors" of A {\displaystyle A} but not eigenvectors in the usual sense. In the absence of (true) eigenvectors, one can look
Spectral_theorem
Method of data analysis
the variance that each eigenvector represents can be calculated by dividing the eigenvalue corresponding to that eigenvector by the sum of all eigenvalues
Principal_component_analysis
of one linearly independent generalized eigenvector of rank 3 (generalized eigenvector rank; see generalized eigenvector), two of rank 2 and four of rank
Modal_matrix
Non-diagonalizable matrix; one lacking a basis of eigenvectors
n} linearly independent eigenvectors. A complete basis is formed by augmenting the eigenvectors with generalized eigenvectors, which are necessary for
Defective_matrix
Measure in graph theory
algorithms that may be used to find this dominant eigenvector. Furthermore, this can be generalized so that the entries in A can be real numbers representing
Eigenvector_centrality
Position in a hierarchy
subset Rank of an elliptic curve Rank of a free module Rank of a generalized eigenvector Rank of a greedoid, the maximal size of a feasible set Rank of
Rank
Function in discrete mathematics
Ding, J. J., Hsue, W. L., & Chang, K. W. (2008). Generalized commuting matrices and their eigenvectors for DFTs, offset DFTs, and other periodic operations
Discrete_Fourier_transform
Description of a quantum-mechanical system
eigenstates, composed of elements outside the Hilbert space, as "generalized eigenvectors". These are used for calculational convenience and do not represent
Schrödinger_equation
Mathematical concept
system, and then compute its eigenvalues and eigenvectors. The eigenvectors (and generalized eigenvectors if they occur) corresponding to eigenvalues with
Center_manifold
Signal processing technique
which are λ = z i {\displaystyle \lambda =z_{i}} . Then, the generalized eigenvectors p i {\displaystyle p_{i}} can be obtained by the following identities:
Generalized pencil-of-function method
Generalized_pencil-of-function_method
Mathematical compact operator
is true for generalized eigenvalues since powers of K − λI and K* − λI are also Fredholm of index 0. Since any generalized λ eigenvector of A is already
Symmetrizable compact operator
Symmetrizable_compact_operator
Linear operator equal to its own adjoint
would say that the eigenvectors are "non-normalizable.") Physicists would then go on to say that these "generalized eigenvectors" form an "orthonormal
Self-adjoint_operator
Property of a mathematical matrix
matrix having as columns the generalized eigenvectors and Λ {\displaystyle \Lambda } is a diagonal matrix of the generalized eigenvalues. Now premultiplication
Definite_matrix
Mathematical entity to describe the probability of each possible measurement on a system
\psi } is a pure state belonging to H {\displaystyle H} , the (generalized) eigenvectors of the position operator do not. Though closely related, pure
Quantum_state
Homomorphisms between simple modules over the same ring are isomorphisms or zero
(z))^{n}m=0} , i.e. if every m ∈ M {\displaystyle m\in M} is a generalized eigenvector of z {\displaystyle z} with eigenvalue χ ( z ) {\displaystyle \chi
Schur's_lemma
equation, w {\displaystyle w} is called a generalized eigenvector and λ {\displaystyle \lambda } is called a generalized eigenvalue. Finding w {\displaystyle
Eigenmoments
Generalized function whose value is zero everywhere except at zero
{\displaystyle P\varphi _{y}=y\varphi _{y}.} That is, φy are the generalized eigenvectors of P. If they form an "orthonormal basis" in the distribution sense
Dirac_delta_function
Notation for quantum states
=\mathbf {r} |\mathbf {r} \rangle .} The position states are "generalized eigenvectors", not elements of the Hilbert space itself, and do not form a countable
Bra–ket_notation
Matrices similar to diagonal matrices
corresponding eigenvalues of T {\displaystyle T} ; with respect to this eigenvector basis, T {\displaystyle T} is represented by D {\displaystyle D} . Diagonalization
Diagonalizable_matrix
Theory of logic to account for observations from quantum theory
operator f(A) is a self-adjoint projection onto the subspace of generalized eigenvectors of A with eigenvalue in [a,b]. That subspace can be interpreted
Quantum_logic
Matrix factorisation in mathematics
T are upper triangular. The generalized Schur decomposition is also sometimes called the QZ decomposition. The generalized eigenvalues λ {\displaystyle
Schur_decomposition
Operator in quantum mechanics
}} ), surjective, endowed with complete families of generalized eigenvectors and real generalized eigenvalues. It is self-adjoint with respect to the
Position_operator
Block diagonal matrix of Jordan blocks
represented by Jordan blocks) of the domain which the associated generalized eigenvectors make a basis for. Let A ∈ M n ( C ) {\displaystyle A\in \mathbb
Jordan_matrix
Mathematical transform
eigenvalues and eigenvectors. Analogously to the classical Fourier transform, the eigenvalues represent frequencies and eigenvectors form what is known
Graph_Fourier_transform
Degree of connectedness within a graph
algorithms that may be used to find this dominant eigenvector. Furthermore, this can be generalized so that the entries in A can be real numbers representing
Centrality
Type of equation involving matrix-valued functions
x r − 1 {\displaystyle x_{0},x_{1},\dots ,x_{r-1}} are called generalized eigenvectors, r {\displaystyle r} is called the length of the Jordan chain,
Nonlinear_eigenproblem
Model for representing text documents
Champion list Compound term processing Conceptual space Eigenvalues and eigenvectors Inverted index Nearest neighbor search Sparse distributed memory w-shingling
Vector_space_model
Concept in mathematics
finding the eigenvectors and eigenvalues of a system A x = λ x {\displaystyle Ax=\lambda x} that is perturbed from one with known eigenvectors and eigenvalues
Eigenvalue_perturbation
Theorem related to ordinary least squares
{k} =(k_{1},\dots ,k_{p+1})^{T}\in \mathbb {R} ^{(p+1)\times 1}} be an eigenvector of H {\displaystyle {\mathcal {H}}} . k ≠ 0 ⟹ ( k 1 v 1 + ⋯ + k p + 1
Gauss–Markov_theorem
{\displaystyle A_{s}} . Constrained generalized inverse Inverse element Moore–Penrose inverse Jordan normal form Generalized eigenvector Drazin, M. P. (1958). "Pseudo-inverses
Drazin_inverse
Construction for adding objects to a Hilbert space
introduced to study spectral theory. They bring together the 'bound state' (eigenvector) and 'continuous spectrum', in one place. Using this notion, a version
Rigged_Hilbert_space
Conjecture on zeros of the zeta function
would also work for the generalized Riemann hypothesis for Dirichlet L-functions. Several results first proved using the generalized Riemann hypothesis were
Riemann_hypothesis
Topics referred to by the same term
using arrows Jordan chain, a sequence of linearly independent generalized eigenvectors of descending rank Markov chain, a discrete-time stochastic process
Chain_(disambiguation)
Construct for Hermitian matrices
{\displaystyle x} is v min {\displaystyle v_{\min }} (the corresponding eigenvector). Similarly, R ( M , x ) ≤ λ max {\displaystyle R(M,x)\leq \lambda _{\max
Rayleigh_quotient
Random process independent of past history
) multiple of a left eigenvector e of the transition matrix P with an eigenvalue of 1. If there is more than one unit eigenvector then a weighted sum of
Markov_chain
Method for finding the exact solution of certain quantum mechanics models
eigenvalues and eigenvectors of the one-dimensional antiferromagnetic isotropic (XXX) Heisenberg model. The approach was later generalized into the quantum
Bethe_ansatz
Sequence of differential equation solutions
L_{n}^{(\alpha )}(x),} which shows that L(α) n is an eigenvector for the eigenvalue n. The generalized Laguerre polynomials are orthogonal over [0, ∞) with
Laguerre_polynomials
Movement of an object which leaves at least one point unchanged
the existence of such a direction is the question of existence of an eigenvector for the matrix A representing the rotation. Every 2D rotation around
Rotation
Construction in functional analysis, useful to solve differential equations
operators, these states are referred to as "generalized eigenvectors" of an observable with "generalized eigenvalues" that do not necessarily belong to
Decomposition of spectrum (functional analysis)
Decomposition_of_spectrum_(functional_analysis)
Polynomial whose roots are the eigenvalues of a matrix
matrix. Eigenvalues and eigenvectors play a fundamental role in linear algebra, since, given a linear transformation, an eigenvector is a vector whose direction
Characteristic_polynomial
American physicist (born 1925)
contracted with the Navy for high-resolution beamforming with generalized eigenvector/eigenvalue (GEVEV) digital signal processing from 1985 through
Gordon_Eugene_Martin
Study of the effects of forces on undeformable bodies
{q}}}}\right),} is the generalized force acting on this one degree of freedom system. If the mechanical system is defined by m generalized coordinates, qj,
Rigid_body_dynamics
and solve a generalized eigenvalue problem. Once eigenvalues and eigenvectors of the linear problem have been determined, eigenvectors and eigenvalues
Quadratic_eigenvalue_problem
Method for finding largest (or smallest) eigenvalues
largest (or smallest) eigenvalues and the corresponding eigenvectors of a symmetric generalized eigenvalue problem A x = λ B x , {\displaystyle Ax=\lambda
LOBPCG
Matrix equal to its conjugate-transpose
like Fourier analysis and signal representation. The eigenvalues and eigenvectors of Hermitian matrices play a crucial role in analyzing signals and extracting
Hermitian_matrix
Set of a matrix's eigenvalues
by matrix multiplication. We now say that x ∈ V is an eigenvector of M if x is an eigenvector of T. Similarly, λ ∈ K is an eigenvalue of M if it is an
Spectrum_of_a_matrix
banded, real symmetric tridiagonal, special real tridiagonal, generalized real, and generalized real symmetric matrices. In addition, it includes subroutines
EISPACK
Non-uniformity of a diffusion process
the corresponding eigenvalues give the magnitude of the peak in each eigenvector direction. FA = 3 2 ( ( λ 1 − λ ^ ) 2 + ( λ 2 − λ ^ ) 2 + ( λ 3 − λ ^
Fractional_anisotropy
subspace is the slow subspace given by the span of the eigenvectors, and generalized eigenvectors, corresponding to the eigenvalue λ = 0 {\displaystyle
Slow_manifold
Matrix decomposition
\mathbf {u} } is a unit length eigenvector of M {\displaystyle \mathbf {M} } . For every unit-length eigenvector v {\displaystyle \mathbf {v} }
Singular_value_decomposition
-representation is useful when introducing the generalized Wannier equation. The Wannier equation can be generalized by including the presence of many electrons
Wannier_equation
Families of matrices in mathematics, physics, and quantum information
particular quantum information, the term generalized Pauli matrices refers to families of matrices which generalize the (linear algebraic) properties of the
Generalizations of Pauli matrices
Generalizations_of_Pauli_matrices
Special kind of square matrix
by using induction on the fact that A has an eigenvector, by taking the quotient space by the eigenvector and inducting to show that A stabilizes a flag
Triangular_matrix
Process of reducing the number of random variables under consideration
correlation) matrix of the data is constructed and the eigenvectors on this matrix are computed. The eigenvectors that correspond to the largest eigenvalues (the
Dimensionality_reduction
Matrix representation of a graph
cut of a graph can be approximated through the Fiedler vector — the eigenvector corresponding to the second smallest eigenvalue of the graph Laplacian
Laplacian_matrix
Study of abstract algebraic structures
but the analysis is much more difficult. Eigenvalues and eigenvectors can be generalized to algebra representations. The generalization of an eigenvalue
Algebra_representation
Mathematical operation
well-known cases, see Eigenvalues and eigenvectors of the second derivative. The second derivative generalizes to higher dimensions through the notion
Second_derivative
Matrix representing a Euclidean rotation
v with (R – I)v = 0, that is Rv = v, a fixed eigenvector. There may also be pairs of fixed eigenvectors in the even-dimensional subspace orthogonal to
Rotation_matrix
is a software library for the parallel computation of eigenvalues and eigenvectors of large, sparse matrices. It can be seen as a module of PETSc that provides
SLEPc
algebra is said to have central character χλ if every vector v is a generalized eigenvector of the center Z with eigenvalue χλ; in other words if z∈Z and v∈V
Translation_functor
Measure of centrality in a network based on nodal influence
between a pair of actors. It is similar to Google's PageRank and to the eigenvector centrality. Katz centrality computes the relative influence of a node
Katz_centrality
Concept in Lie algebra representation theory
a basis of V consisting of simultaneous eigenvectors of all elements of S. Each of these common eigenvectors v ∈ V defines a linear functional on the
Weight (representation theory)
Weight_(representation_theory)
Probability distribution and special case of gamma distribution
{\displaystyle {\bar {1}}} the all ones vector. M {\displaystyle M} has one eigenvector b 1 := 1 n 1 ¯ {\displaystyle b_{1}:={\textstyle {\frac {1}{\sqrt {n}}}}{\bar
Chi-squared_distribution
Array of numbers
expressible as a linear combination of eigenvectors. In both cases, all eigenvalues are real. This theorem can be generalized to infinite-dimensional situations
Matrix_(mathematics)
Function used in signal processing
values of N) to L × σt for σt < 0.14. A more generalized version of the Gaussian window is the generalized normal window. Retaining the notation from the
Window_function
Coordinate transformation that preserves the form of Hamilton's equations
is much broader, since the old generalized coordinates, momenta and even time may be combined to form the new generalized coordinates and momenta. Canonical
Canonical_transformation
the eigenvector, and its length is the eigenvalue. All other eigenvectors will be perpendicular to this. The minimax principle also generalizes to eigenvalues
Courant_minimax_principle
Collection of mathematical theories
mathematics, spectral theory is an inclusive term for theories extending the eigenvector and eigenvalue theory of a single square matrix to a much broader theory
Spectral_theory
Numerical linear algebra algorithm
algorithm is an iterative method for the calculation of the eigenvalues and eigenvectors of a real symmetric matrix (a process known as diagonalization). It is
Jacobi_eigenvalue_algorithm
Iterative method for approximating eigenvectors
iterative method. Arnoldi finds an approximation to the eigenvalues and eigenvectors of general (possibly non-Hermitian) matrices by constructing an orthonormal
Arnoldi_iteration
Mathematical inequality relating inner products and norms
{u} } is an eigenvector of A 2 {\displaystyle A^{2}} . From here it is straightforward to deduce that A {\displaystyle A} has an eigenvector, then the spectral
Cauchy–Schwarz_inequality
Representation of a quantum mechanical system
\rangle } invariant must have | ψ ⟩ {\displaystyle |\psi \rangle } as an eigenvector. Since the corresponding eigenvalue must be a complex number of modulus
Bloch_sphere
Singularities in the parameter space
in the parameter space where two or more eigenstates (eigenvalues and eigenvectors) coalesce. These points appear in dissipative systems, which make the
Exceptional_point
Python library for graphs and networks
come from the third eigenvector. Scale and center the resulting layout as needed. Nodes in dense clusters have similar eigenvector entries, causing them
NetworkX
Matrix that commutes with its conjugate transpose
and the columns of U are the eigenvectors of A. The matching eigenvalues in Λ come in the same order as the eigenvectors are ordered as columns of U.
Normal_matrix
Calculation rule in quantum mechanics
{\displaystyle \lambda _{i}} is one-dimensional and spanned by the normalized eigenvector | λ i ⟩ {\displaystyle |\lambda _{i}\rangle } , P i {\displaystyle P_{i}}
Born_rule
Matrix whose conjugate transpose is its negative (additive inverse)
skew-Hermitian matrices are normal. Hence they are diagonalizable and their eigenvectors for distinct eigenvalues must be orthogonal. All entries on the main
Skew-Hermitian_matrix
Non-self-adjoint compact operator used to solve boundary value problems for the Laplacian
each of the statements for either T or T*. To check that T has no generalized eigenvectors with eigenvalue 1/2 it suffices to show that T K φ − 1 2 φ = 1
Neumann–Poincaré_operator
Two-dimensional conformal field theory
is called the coupling constant. In a free field theory, the energy eigenvectors e 2 α φ {\displaystyle e^{2\alpha \varphi }} are linearly independent
Liouville_field_theory
Branch of mathematics
If f is a linear endomorphism of a vector space V over a field F, an eigenvector of f is a nonzero vector v of V such that f(v) = av for some scalar a
Linear_algebra
Functional analysis concept
the existence of one eigenvector x {\displaystyle x} of T {\displaystyle T} . In finite dimension, the existence of an eigenvector can be shown in (at
Compact operator on Hilbert space
Compact_operator_on_Hilbert_space
Interaction of a quantum system with a classical observer
spectral theory; the present article will avoid them whenever possible. The eigenvectors of a von Neumann observable form an orthonormal basis for the Hilbert
Measurement in quantum mechanics
Measurement_in_quantum_mechanics
Representation theory of groups
simultaneous eigenvectors for all the n×n circulants. In fact if ζ is any n-th root of unity, the element 1 + ζg + ζ2g2 + ... + ζn−1gn−1 is an eigenvector for
Regular_representation
Iterative procedure
procedure used to fit a generalized additive model. It was introduced in 1985 by Leo Breiman and Jerome Friedman along with generalized additive models. In
Backfitting_algorithm
Set of related ordination techniques used in information visualization
{\textstyle \lambda _{1},\lambda _{2},...,\lambda _{m}} and corresponding eigenvectors e 1 , e 2 , . . . , e m {\textstyle e_{1},e_{2},...,e_{m}} of B {\textstyle
Multidimensional_scaling
Dimensionality reduction algorithm
{\displaystyle y} is an eigenvector of S {\displaystyle S} , then V 1 N − 1 y {\displaystyle V_{1}^{N-1}y} is an approximate eigenvector of A {\displaystyle
Dynamic_mode_decomposition
number of pixels in the image. Since only one eigenvector, corresponding to the second smallest generalized eigenvalue, is used by the uncut algorithm,
Segmentation-based object categorization
Segmentation-based_object_categorization
Model of how neurons in the brain or artificial neural networks learn over time
is a single-neuron special case of the Generalized Hebbian Algorithm. However, Oja's rule can also be generalized in other ways to varying degrees of stability
Oja's_rule
Probability theory concept
i {\displaystyle i} -th column is the eigenvector v i {\displaystyle \,v_{i}} . Note that since the eigenvectors are linearly independent, the matrix P
Fractional_Brownian_motion
Neuroscientific theory
^{*}\mathbf {x} } Because, again, c ∗ {\displaystyle \mathbf {c} ^{*}} is the eigenvector corresponding to the largest eigenvalue of the correlation matrix between
Hebbian_theory
Way of inferring information from cross-covariance matrices
the maximum of correlation is attained if c {\displaystyle c} is the eigenvector with the maximum eigenvalue for the matrix Σ X X − 1 / 2 Σ X Y Σ Y Y
Canonical_correlation
Matrix with the same number of rows and columns
eigenbasis; i.e., every vector is expressible as a linear combination of eigenvectors. In both cases, all eigenvalues are real. A symmetric n×n-matrix is called
Square_matrix
Mathematical function
matrix C {\displaystyle C} and changing the integration variables to the eigenvectors of C {\displaystyle C} . More generally a shifted Gaussian function is
Gaussian_function
GENERALIZED EIGENVECTOR
GENERALIZED EIGENVECTOR
Surname or Lastname
English
English : status name from Middle English squyer ‘esquire’, ‘a man belonging to the feudal rank immediately below that of knight’ (from Old French esquier ‘shield bearer’). At first it denoted a young man of good birth attendant on a knight, or by extension any attendant or servant, but by the 14th century the meaning had been generalized, and referred to social status rather than age. By the 17th century, the term denoted any member of the landed gentry, but this is unlikely to have influenced the development of the surname.
GENERALIZED EIGENVECTOR
GENERALIZED EIGENVECTOR
Boy/Male
Indian, Punjabi, Sikh
Loving / Joyful
Girl/Female
Hindu, Indian
Prayer
Boy/Male
Tamil
The first Lord or Lord Vishnu
Girl/Female
Tamil
Soft spoken
Girl/Female
Hindu
Flower
Boy/Male
German
Strong Judgment
Boy/Male
Hebrew
Judge. Biblical fifth son of Jacob and founder of one of the twelve tribes of Israel. An...
Girl/Female
Sikh
Love for coast
Boy/Male
Indian
This ws the name of Wahb Ibn
Male
Egyptian
, Asshur-nadin, ("Asshur gives").
GENERALIZED EIGENVECTOR
GENERALIZED EIGENVECTOR
GENERALIZED EIGENVECTOR
GENERALIZED EIGENVECTOR
GENERALIZED EIGENVECTOR
v. t.
To impregnate with a mineral; as, mineralized water.
v. t.
To bring under a genus or under genera; to view in relation to a genus or to genera.
v. t.
To make universal; to generalize.
a.
Comprising structural characters which are separated in more specialized forms; synthetic; as, a generalized type.
a.
Capable of being generalized, or reduced to a general form of statement, or brought under a general rule.
n.
The act or process of centralizing, or the state of being centralized; the act or process of combining or reducing several parts into a whole; as, the centralization of power in the general government; the centralization of commerce in a city.
imp. & p. p.
of Generalize
v. t.
To derive or deduce (a general conception, or a general principle) from particulars.
n.
A fishlike creature (Amphioxus lanceolatus), two or three inches long, found in temperature seas; -- also called the lancelet. Its body is pointed at both ends. It is the lowest and most generalized of the vertebrates, having neither brain, skull, vertebrae, nor red blood. It forms the type of the group Acrania, Leptocardia, etc.
n.
A generalized concept of magnitude.
imp. & p. p.
of Federalize
v. t.
To generalize or conclude as an inference from all the particulars; -- the opposite of deduce.
v. i.
To form into a genus; to view objects in their relations to a genus or class; to take general or comprehensive views.
imp. & p. p.
of Centralize
imp. & p. p.
of Mineralize
n.
One who takes general or comprehensive views.
p. pr. & vb. n.
of Generalize
n.
The system by which power is centralized, as in a government.
v. t.
To apply to other genera or classes; to use with a more extensive application; to extend so as to include all special cases; to make universal in application, as a formula or rule.