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Using the Leibniz formula for determinants, the left-hand side of equation is a polynomial function of the variable λ and the degree of this polynomial is n, the order of the matrix A. Its coefficients depend on the entries of A, except that its term of degree n is always (−1) n λ n. This polynomial is called the characteristic polynomial of A.
Notation: The index j represents the jth eigenvalue or eigenvector. The index i represents the ith component of an eigenvector. Both i and j go from 1 to n, where the matrix is size n x n. Eigenvectors are normalized. The eigenvalues are ordered in descending order.
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.
This differential equation can be solved by multiplying both sides by () and integrating. Its solution, the exponential function =, is the eigenfunction of the derivative operator, where f 0 is a parameter that depends on the boundary conditions. Note that in this case the eigenfunction is itself a function of its associated eigenvalue λ ...
Quadratic eigenvalue problems arise naturally in the solution of systems of second order linear differential equations without forcing: ″ + ′ + = Where (), and ,,.If all quadratic eigenvalues of () = + + are distinct, then the solution can be written in terms of the quadratic eigenvalues and right quadratic eigenvectors as
The number is known as the (nonlinear) eigenvalue, the vector as the (nonlinear) eigenvector, and (,) as the eigenpair. The matrix M ( λ ) {\displaystyle M(\lambda )} is singular at an eigenvalue λ {\displaystyle \lambda } .
The main application of the method is the situation when an approximation to an eigenvalue is found and one needs to find the corresponding approximate eigenvector. In such a situation the inverse iteration is the main and probably the only method to use.