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 . Overview and definition 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 . Diagonalizable matrices are of particular interest since matrix functions of them can be computed easily. On the other hand, if does not have linearly independent eigenvectors associated with it, then is not diagonalizable. Definition: A vector is a generalized eigenvector of rank m of the matrix and corresponding to the eigenvalue if but Clearly, a generalized eigenvector of rank 1 is an