; Example 3 The matrix has an eigenvalue of algebraic multiplicity and an eigenvalue of algebraic multiplicity . We also have . For we have . The first integer for which has rank is . We now define Consequently, there will be three linearly independent generalized eigenvectors; one each of ranks 3, 2 and 1. Since corresponds to a single chain of three linearly independent generalized eigenvectors, we know that there is a generalized eigenvector of rank 3 corresponding to such that but Equations () and () represent linear systems that can be solved for . Let Then and Thus, in order to satisfy the conditions () and (), we must have and . No restrictions are placed on and . By choosing , we obtain as a generalized eigenvector of rank 3 corresponding to . Note that it is possible to obtain infinitely many other generalized eigenvectors of rank 3 by choosing different values of , and , with . Our first choice, however, is the simplest. Now using equations (), we obtain and as generalized eigenvectors of rank 2 and 1, respectively, where and The simple eigenvalue can be dealt with using standard techniques and has an ordinary eigenvector A canonical basis for is and are generalized eigenvectors associated with , while is the ordinary eigenvector associated with . This is a fairly simple example. In general, the numbers of linearly independent generalized eigenvectors of rank will not always be equal. That is, there may be several chains of different lengths corresponding to a particular eigenvalue. Generalized modal matrix Let be an n × n matrix. A generalized modal matrix for is an n × n matrix whose columns, considered as vectors, form a canonical basis for and appear in according to the following rules: All Jordan chains consisting of one vector (that is, one vector in length) appear in the first columns of . All vectors of one chain appear together in adjacent columns of . Each chain appears in in order of increasing rank (that is, the generalized eigenvector of rank 1 appears before the generalized eigenvector of rank 2 of the same chain, which appears before the generalized eigenvector of rank 3 of the same chain, etc.). Jordan normal form Let be an n-dimensional vector space; let be a linear map in , the set of all linear maps from into itself; and let be