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Concept
Generalized eigenvector
Summary
In linear algebra, a generalized eigenvector of an matrix is a vector which satisfies certain criteria which are more relaxed than those for an (ordinary) eigenvector.
Let be an -dimensional vector space and let be the matrix representation of a linear map from to with respect to some ordered basis.
There may not always exist a full set of linearly independent eigenvectors of that form a complete basis for . That is, the matrix may not be diagonalizable. This happens when the algebraic multiplicity of at least one eigenvalue is greater than its geometric multiplicity (the nullity of the matrix , or the dimension of its nullspace). In this case, is called a defective eigenvalue and is called a defective matrix.
A generalized eigenvector corresponding to , together with the matrix generate a Jordan chain of linearly independent generalized eigenvectors which form a basis for an invariant subspace of .
Using generalized eigenvectors, a set of linearly independent eigenvectors of can be extended, if necessary, to a complete basis for . This basis can be used to determine an "almost diagonal matrix" in Jordan normal form, similar to , which is useful in computing certain matrix functions of . The matrix is also useful in solving the system of linear differential equations where need not be diagonalizable.
The dimension of the generalized eigenspace corresponding to a given eigenvalue is the algebraic multiplicity of .
There are several equivalent ways to define an ordinary eigenvector. For our purposes, an eigenvector associated with an eigenvalue of an × matrix is a nonzero vector for which , where is the × identity matrix and is the zero vector of length . That is, is in the kernel of the transformation . If has linearly independent eigenvectors, then is similar to a diagonal matrix . That is, there exists an invertible matrix such that is diagonalizable through the similarity transformation . The matrix is called a spectral matrix for . The matrix is called a modal matrix for .
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