Iwahori-Hecke algebra has the standard basis , the group is partially ordered by the Bruhat order which is interval finite and has a dualization operation defined by . This is a precanonical structure on that satisfies the sufficient condition above and the corresponding canonical basis of is the Kazhdan–Lusztig basis with being the Kazhdan–Lusztig polynomials. Linear algebra If we are given an n × n matrix and wish to find a matrix in Jordan normal form, similar to , we are interested only in sets of linearly independent generalized eigenvectors. A matrix in Jordan normal form is an "almost diagonal matrix," that is, as close to diagonal as possible. A diagonal matrix is a special case of a matrix in Jordan normal form. An ordinary eigenvector is a special case of a generalized eigenvector. Every n × n matrix possesses n linearly independent generalized eigenvectors. Generalized eigenvectors corresponding to distinct eigenvalues are linearly independent. If is an eigenvalue of of algebraic multiplicity , then will have linearly independent generalized eigenvectors corresponding to . For any given n × n matrix , there are infinitely many ways to pick the n linearly independent generalized eigenvectors. If they are chosen in a particularly judicious manner, we can use these vectors to show that is similar to a matrix in Jordan normal form. In particular, 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. Computation 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'' (generalized eigenvector rank; see generalized eigenvector) corresponding to the eigenvalue that will appear in a canonical basis for . Note that Once we have determined the number of generalized eigenvectors of each rank that a canonical basis has, we can obtain the vectors explicitly (see generalized eigenvector). Example This example illustrates a canonical basis