Search results
Results from the WOW.Com Content Network
For a matrix, eigenvalues and eigenvectors can be used to decompose the matrix—for example by diagonalizing it. Eigenvalues and eigenvectors give rise to many closely related mathematical concepts, and the prefix eigen-is applied liberally when naming them:
The eigenvalues are real. The eigenvectors of A −1 are the same as the eigenvectors of A. Eigenvectors are only defined up to a multiplicative constant. That is, if Av = λv then cv is also an eigenvector for any scalar c ≠ 0. In particular, −v and e iθ v (for any θ) are also eigenvectors.
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 ...
It is used in all applications that involve approximating eigenvalues and eigenvectors, often under different names. In quantum mechanics , where a system of particles is described using a Hamiltonian , the Ritz method uses trial wave functions to approximate the ground state eigenfunction with the lowest energy.
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.
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 x 0 i , λ 0 i , i = 1 , … n {\displaystyle x_{0i},\lambda _{0i ...
In mathematics, power iteration (also known as the power method) is an eigenvalue algorithm: given a diagonalizable matrix, the algorithm will produce a number , which is the greatest (in absolute value) eigenvalue of , and a nonzero vector , which is a corresponding eigenvector of , that is, =.
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.