the matrix representation of with respect to some ordered basis. It can be shown that if the characteristic polynomial of factors into linear factors, so that has the form where are the distinct eigenvalues of , then each is the algebraic multiplicity of its corresponding eigenvalue and is similar to a matrix in Jordan normal form, where each appears consecutive times on the diagonal, and the entry directly above each (that is, on the superdiagonal) is either 0 or 1. All other entries (that is, off the diagonal and superdiagonal) are 0. More precisely, is a Jordan matrix whose Jordan blocks corresponding to the same eigenvalue are grouped together (but no ordering is imposed among the eigenvalues, nor among the blocks for a given eigenvalue). The matrix is as close as one can come to a diagonalization of . If is diagonalizable, then all entries above the diagonal are zero. Note that some textbooks have the ones on the subdiagonal, that is, immediately below the main diagonal instead of on the superdiagonal. The eigenvalues are still on the main diagonal. Every n × n matrix is similar to a matrix in Jordan normal form, obtained through the similarity transformation , where is a generalized modal matrix for . (See Note above.) Example 4 Find a matrix in Jordan normal form that is similar to Solution: The characteristic equation of is , hence, is an eigenvalue of algebraic multiplicity three. Following the procedures of the previous sections, we find that and Thus, and , which implies that a canonical basis for will contain one linearly independent generalized eigenvector of rank 2 and two linearly independent generalized eigenvectors of rank 1, or equivalently, one chain of two vectors and one chain of one vector . Designating , we find that and where is a generalized modal matrix for , the columns of are a canonical basis for , and . Note that since generalized eigenvectors themselves are not unique, and since some of the columns of both and may be interchanged, it follows that both and are not unique. Example 5 In Example 3, we found a canonical basis of linearly independent generalized eigenvectors for a matrix . A generalized modal matrix for is A matrix in Jordan normal form, similar to is so that . Applications Matrix functions Three of the most fundamental operations which can be performed on square matrices