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Eigenvalues and eigenvectors. In linear algebra, an eigenvector (/ ˈaɪɡən -/ EYE-gən-) or characteristic vector is a vector that has its direction unchanged by a given linear transformation. More precisely, an eigenvector, , of a linear transformation, , is scaled by a constant factor, , when the linear transformation is applied to it: .
In mathematics, an eigenvalue perturbation problem is that of finding the eigenvectors and eigenvalues of a system that is perturbed from one with known eigenvectors and eigenvalues . This is useful for studying how sensitive the original system's eigenvectors and eigenvalues are to changes in the system. This type of analysis was popularized ...
Eigendecomposition of a matrix. In linear algebra, eigendecomposition is the factorization of a matrix into a canonical form, whereby the matrix is represented in terms of its eigenvalues and eigenvectors. Only diagonalizable matrices can be factorized in this way. When the matrix being factorized is a normal or real symmetric matrix, the ...
Quadratic eigenvalue problem. In mathematics, the quadratic eigenvalue problem[1] (QEP), is to find scalar eigenvalues , left eigenvectors and right eigenvectors such that. where , with matrix coefficients and we require that , (so that we have a nonzero leading coefficient). There are eigenvalues that may be infinite or finite, and possibly zero.
In linear algebra, a generalized eigenvector of an matrix is a vector which satisfies certain criteria which are more relaxed than those for an (ordinary) eigenvector. [1] Let be an -dimensional vector space and let be the matrix representation of a linear map from to with respect to some ordered basis.
Divide-and-conquer eigenvalue algorithms are a class of eigenvalue algorithms for Hermitian or real symmetric matrices that have recently (circa 1990s) become competitive in terms of stability and efficiency with more traditional algorithms such as the QR algorithm. The basic concept behind these algorithms is the divide-and-conquer approach ...
Given an n × n square matrix A of real or complex numbers, an eigenvalue λ and its associated generalized eigenvector v are a pair obeying the relation [1] =,where v is a nonzero n × 1 column vector, I is the n × n identity matrix, k is a positive integer, and both λ and v are allowed to be complex even when A is real.l When k = 1, the vector is called simply an eigenvector, and the pair ...
As stated in the introduction, for any vector x, one has (,) [,], where , are respectively the smallest and largest eigenvalues of .This is immediate after observing that the Rayleigh quotient is a weighted average of eigenvalues of M: (,) = = = = where (,) is the -th eigenpair after orthonormalization and = is the th coordinate of x in the eigenbasis.