Eigenvalue algorithm

In numerical analysis, one of the most important problems is designing efficient and stable algorithms for finding the eigenvalues of a matrix. These eigenvalue algorithms may also find eigenvectors.

Given an $n \times n$ square matrix $A$ of real or complex numbers, an eigenvalue $λ$ and its associated generalized eigenvector $v$ are a pair obeying the relation

where $v$ is a nonzero $n \times 1$ column vector, $I$ is the $n \times n$ identity matrix, $k$ is a positive integer, and both $λ$ and $v$ are allowed to be complex even when $A$ is real. When $k = 1$ , the vector is called simply an eigenvector, and the pair is called an eigenpair. In this case, $A v = λ v$ . Any eigenvalue $λ$ of $A$ has ordinary eigenvectors associated to it, for if $k$ is the smallest integer such that $(A - λ I) k v = 0$ for a generalized eigenvector $v$ , then $(A - λ I) k -1 v$ is an ordinary eigenvector. The value $k$ can always be taken as less than or equal to $n$ . In particular, $(A - λ I) n v = 0$ for all generalized eigenvectors $v$ associated with $λ.$

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