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Concept
Eigenvalue algorithm
Summary
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.
Eigenvalues and eigenvectors and Generalized eigenvector
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
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. When k = 1, the vector is called simply an eigenvector, and the pair is called an eigenpair. In this case, Av = λv. Any eigenvalue λ of A has ordinary eigenvectors associated to it, for if k is the smallest integer such that (A − λI)k v = 0 for a generalized eigenvector v, then (A − λI)k−1 v is an ordinary eigenvector. The value k can always be taken as less than or equal to n. In particular, (A − λI)n v = 0 for all generalized eigenvectors v associated with λ.
For each eigenvalue λ of A, the kernel ker(A − λI) consists of all eigenvectors associated with λ (along with 0), called the eigenspace of λ, while the vector space ker((A − λI)n) consists of all generalized eigenvectors, and is called the generalized eigenspace. The geometric multiplicity of λ is the dimension of its eigenspace. The algebraic multiplicity of λ is the dimension of its generalized eigenspace. The latter terminology is justified by the equation
where det is the determinant function, the λi are all the distinct eigenvalues of A and the αi are the corresponding algebraic multiplicities. The function pA(z) is the characteristic polynomial of A. So the algebraic multiplicity is the multiplicity of the eigenvalue as a zero of the characteristic polynomial. Since any eigenvector is also a generalized eigenvector, the geometric multiplicity is less than or equal to the algebraic multiplicity. The algebraic multiplicities sum up to n, the degree of the characteristic polynomial.
Official source
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