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The corresponding eigenvalue, characteristic value, or characteristic root is the multiplying factor (possibly negative). Geometrically, vectors are multi-dimensional quantities with magnitude and direction, often pictured as arrows. A linear transformation rotates, stretches, or shears the vectors upon which it acts. A linear transformation's ...
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 ...
Rayleigh quotient iteration is an eigenvalue algorithm which extends the idea of the inverse iteration by using the Rayleigh quotient to obtain increasingly accurate eigenvalue estimates. Rayleigh quotient iteration is an iterative method, that is, it delivers a sequence of approximate solutions that converges to a true solution in the limit ...
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.
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 numerical linear algebra, the Jacobi eigenvalue algorithm is an iterative method for the calculation of the eigenvalues and eigenvectors of a real symmetric matrix ...
This solution of the vibrating drum problem is, at any point in time, an eigenfunction of the Laplace operator on a disk.. In mathematics, an eigenfunction of a linear operator D defined on some function space is any non-zero function in that space that, when acted upon by D, is only multiplied by some scaling factor called an eigenvalue.
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. Then, the Rayleigh theorem for eigenvalues states that λ ′ i ≤ λ i for i = 1 to N.