Search results
Results from the WOW.Com Content Network
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
Comparing this equation to equation , it follows immediately that a left eigenvector of is the same as the transpose of a right eigenvector of , with the same eigenvalue. Furthermore, since the characteristic polynomial of A T {\displaystyle A^{\textsf {T}}} is the same as the characteristic polynomial of A {\displaystyle A} , the left and ...
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.
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.
The row vectors of are the left eigenvectors of . When a complex matrix A ∈ C n × n {\displaystyle A\in \mathbb {C} ^{n\times n}} is a Hermitian matrix (or more generally a normal matrix ), eigenvectors of A {\displaystyle A} can be chosen to form an orthonormal basis of C n {\displaystyle \mathbb {C} ^{n}} , and P {\displaystyle P} can be ...
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 } .
In numerical linear algebra, the Arnoldi iteration is an eigenvalue algorithm and an important example of an iterative method.Arnoldi finds an approximation to the eigenvalues and eigenvectors of general (possibly non-Hermitian) matrices by constructing an orthonormal basis of the Krylov subspace, which makes it particularly useful when dealing with large sparse matrices.