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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 ...
Another method can extend Givens rotations to complex matrices. A diagonal matrix whose diagonal elements have unit magnitudes but arbitrary phases is unitary. Let A be a matrix for which it is desired to make the ji element be zero using the rows and columns i and j>i.
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 (a process known as diagonalization).
Given an eigenvalue λ, consider the set = {: =}, which is the union of the zero vector with the set of all eigenvectors associated with λ . E is called the eigenspace or characteristic space of T associated with λ .
More generally, we can factor a complex m×n matrix A, with m ≥ n, as the product of an m×m unitary matrix Q and an m×n upper triangular matrix R.As the bottom (m−n) rows of an m×n upper triangular matrix consist entirely of zeroes, it is often useful to partition R, or both R and Q:
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.
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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.