In mathematics, a canonical basis is a basis of an algebraic structure that is canonical in a sense that depends on the precise context:
In representation theory there are several basis that are called "canonical", e.g. Lusztig's canonical basis and closely related Kashiwara's crystal basis in quantum groups and their representations. There is a general concept underlying these basis:
Consider the ring of integral Laurent polynomials with its two subrings and the automorphism that is defined by .
A precanonical structure on a free -module consists of
- A standard basis of ,
- A partial order on that is interval finite, i.e. is finite for all ,
- A dualization operation, i.e. a bijection of order two that is -semilinear and will be denoted by as well.
If a precanonical structure is given, then one can define the submodule of .
A canonical basis at of the precanonical structure is then a -basis of that satisfies:
for all . A canonical basis at is analogously defined to be a basis that satisfies
for all . The naming "at " alludes to the fact and hence the "specialization" corresponds to quotienting out the relation .
One can show that there exists at most one canonical basis at v = 0 (and at most one at ) for each precanonical structure. A sufficient condition for existence is that the polynomials defined by satisfy and .
A canonical basis at v = 0 () induces an isomorphism from to ( respectively).
The canonical basis of quantum groups in the sense of Lusztig and Kashiwara are canonical basis at .
Let be a Coxeter group. The corresponding 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 at is the Kazhdan–Lusztig basis
with being the Kazhdan–Lusztig polynomials.
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.
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).
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).
This example illustrates a canonical basis 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.
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.
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 .
- Bronson, Richard (1970), Matrix Methods: An Introduction, New York: Academic Press, LCCN 70097490
- Deng, Bangming; Ju, Jie; Parshall, Brian; Wang, Jianpan (2008), Finite Dimensional Algebras and Quantum Groups, Mathematical surveys and monographs, 150, Providence, R.I.: American Mathematical Society, ISBN 9780821875315
- Nering, Evar D. (1970), Linear Algebra and Matrix Theory (2nd ed.), New York: Wiley, LCCN 76091646