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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 ...
The index j represents the jth eigenvalue or eigenvector and runs from 1 to . Assuming the equation is defined on the domain [,], the following are the eigenvalues and normalized eigenvectors. The eigenvalues are ordered in descending order.
This equation is called the eigenvalue equation for T, and the scalar λ is the eigenvalue of T corresponding to the eigenvector v. T(v) is the result of applying the transformation T to the vector v, while λv is the product of the scalar λ with v. [37] [38]
Once the eigenvalues are computed, the eigenvectors could be calculated by solving the equation (), = using Gaussian elimination or any other method for solving matrix equations. However, in practical large-scale eigenvalue methods, the eigenvectors are usually computed in other ways, as a byproduct of the eigenvalue computation.
In matrix theory, Sylvester's formula or Sylvester's matrix theorem (named after J. J. Sylvester) or Lagrange−Sylvester interpolation expresses an analytic function f(A) of a matrix A as a polynomial in A, in terms of the eigenvalues and eigenvectors of A. [1] [2] It states that [3]
According to S. Ilanko, [2] citing Richard Courant, both Lord Rayleigh and Walther Ritz independently conceived the idea of utilizing the equivalence between boundary value problems of partial differential equations on the one hand and problems of the calculus of variations on the other hand for numerical calculation of the solutions, by ...
Let the resulting eigenvalues be ordered from the smallest (lowest), λ 1, to the largest (highest), λ N. Let the same eigenvalue equation be solved using a basis set of dimension N + 1 that comprises the previous N functions plus an additional one. Let the resulting eigenvalues be ordered from the smallest, λ ′ 1, to the largest, λ ′ N+1.
Here it is assumed that floating point operations are optimally rounded to the nearest floating point number. 2. The upper triangle of the matrix S is destroyed while the lower triangle and the diagonal are unchanged. Thus it is possible to restore S if necessary according to for k := 1 to n−1 do !