Vector satisfying some of the criteria of an eigenvector
Not to be confused with Generalized eigenvalue problem.
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.[1]
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.[2][3] 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.[4]
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 .[5][6][7]
Using generalized eigenvectors, a set of linearly independent eigenvectors of can be extended, if necessary, to a complete basis for .[8] 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 .[9] The matrix is also useful in solving the system of linear differential equations where need not be diagonalizable.[10][11]
The dimension of the generalized eigenspace corresponding to a given eigenvalue is the algebraic multiplicity of .[12]
^Bronson (1970, p. 189)
^Beauregard & Fraleigh (1973, p. 310)
^Nering (1970, p. 118)
^Golub & Van Loan (1996, p. 316)
^Beauregard & Fraleigh (1973, p. 319)
^Bronson (1970, pp. 194–195)
^Golub & Van Loan (1996, p. 311)
^Bronson (1970, p. 196)
^Bronson (1970, p. 189)
^Beauregard & Fraleigh (1973, pp. 316–318)
^Nering (1970, p. 118)
^Bronson (1970, p. 196)
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