with two Jordan chains. 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 . For we have has rank 5, has rank 4, has rank 3, has rank 2. Therefore Thus, a canonical basis for will have, corresponding to one generalized eigenvector each of ranks 4, 3, 2 and 1. For we have has rank 5, has rank 4. Therefore Thus, a canonical basis for will have, corresponding to one generalized eigenvector each of ranks 2 and 1. A canonical basis for is is the ordinary eigenvector associated with . and are generalized eigenvectors associated with . is the ordinary eigenvector associated with . is a generalized eigenvector associated with . A matrix in Jordan normal form, similar to is obtained as follows: where the matrix is a generalized modal matrix for and . See also Canonical form Change of basis Normal basis Normal form (disambiguation) Polynomial basis Notes References Linear algebra Abstract algebra Lie algebras Representation theory Quantum groups