Search results
Results from the WOW.Com Content Network
The set of all eigenvectors of a linear transformation, each paired with its corresponding eigenvalue, is called the eigensystem of that transformation. [7] [8] The set of all eigenvectors of T corresponding to the same eigenvalue, together with the zero vector, is called an eigenspace, or the characteristic space of T associated with that ...
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 ...
Let A be a square n × n matrix with n linearly independent eigenvectors q i (where i = 1, ..., n).Then A can be factored as = where Q is the square n × n matrix whose i th column is the eigenvector q i of A, and Λ is the diagonal matrix whose diagonal elements are the corresponding eigenvalues, Λ ii = λ i.
Using generalized eigenvectors, a set of linearly independent eigenvectors of can be extended, if necessary, to a complete basis for . [8] This basis can be used to determine an "almost diagonal matrix" J {\displaystyle J} in Jordan normal form , similar to A {\displaystyle A} , which is useful in computing certain matrix functions of A ...
The Lanczos algorithm is most often brought up in the context of finding the eigenvalues and eigenvectors of a matrix, but whereas an ordinary diagonalization of a matrix would make eigenvectors and eigenvalues apparent from inspection, the same is not true for the tridiagonalization performed by the Lanczos algorithm; nontrivial additional steps are needed to compute even a single eigenvalue ...
Let us take = [], then = [] with eigenvalues , and the corresponding eigenvectors = = [], = = [], so that the Ritz values are , and the Ritz vectors are ~ ~ = = [], ~ ~ = = []. We observe that each one of the Ritz vectors is exactly one of the eigenvectors of A {\displaystyle A} for the given V {\displaystyle V} as well as the Ritz values give ...
When the eigenvalues (and eigenvectors) of a symmetric matrix are known, the following values are easily calculated. Singular values The singular values of a (square) matrix A {\displaystyle A} are the square roots of the (non-negative) eigenvalues of A T A {\displaystyle A^{T}A} .
In numerical analysis, inverse iteration (also known as the inverse power method) is an iterative eigenvalue algorithm.It allows one to find an approximate eigenvector when an approximation to a corresponding eigenvalue is already known.