This example is more complex than Example 1. Unfortunately, it is a little difficult to construct an interesting example of low order. The matrix has eigenvalues and with algebraic multiplicities and , but geometric multiplicities and . The generalized eigenspaces of are calculated below. is the ordinary eigenvector associated with . is a generalized eigenvector associated with . is the ordinary eigenvector associated with . and are generalized eigenvectors associated with . This results in a basis for each of the generalized eigenspaces of . Together the two chains of generalized eigenvectors span the space of all 5-dimensional column vectors. An "almost diagonal" matrix in Jordan normal form, similar to is obtained as follows: where is a generalized modal matrix for , the columns of are a canonical basis for , and . Jordan chains Definition: Let be a generalized eigenvector of rank m corresponding to the matrix and the eigenvalue . The chain generated by is a set of vectors given by Thus, in general, The vector , given by (), is a generalized eigenvector of rank j corresponding to the eigenvalue . A chain is a linearly independent set of vectors. Canonical basis Definition: A set of n linearly independent generalized eigenvectors is a canonical basis if it is composed entirely of Jordan chains. Thus, once we have determined that a generalized eigenvector of rank m is in a canonical basis, it follows that the m − 1 vectors that are in the Jordan chain generated by are also in the canonical basis. Let be an eigenvalue of of algebraic multiplicity . First, find the ranks (matrix ranks) of the matrices . The integer is determined to be the first integer for which has rank (n being the number of rows or columns of , that is, is n × n). Now define The variable designates the number of linearly independent generalized eigenvectors of rank k corresponding to the eigenvalue that will appear in a canonical basis for . Note that . Computation of generalized eigenvectors In the preceding sections we have seen techniques for obtaining the linearly independent generalized eigenvectors of a canonical basis for the vector space associated with an matrix . These techniques can be combined into a procedure: Solve the characteristic equation of for eigenvalues and their algebraic multiplicities ; For each Determine ; Determine ; Determine for ; Determine each Jordan chain for