Two linear transformations each tri-diagonal with respect to an eigenbasis of the other; the TD-D canonical form and the LB-UB canonical form
| dc.creator | Terwilliger, Paul M. | |
| dc.date | 2003-04-06 | |
| dc.date.accessioned | 2026-07-07T04:56:39Z | |
| dc.date.available | 2026-07-07T04:56:39Z | |
| dc.description | Let $\K$ denote a field and let $V$ denote a vector space over $\K$ with finite positive dimension. We consider an ordered pair of linear transformations $A:V\to V$ and $B:V\to V$ which satisfy both (i), (ii) below. (i) There exists a basis for $V$ with respect to which the matrix representing $A$ is irreducible tridiagonal and the matrix representing $B$ is diagonal; (ii) There exists a basis for $V$ with respect to which the matrix representing $A$ is diagonal and the matrix representing $B$ is irreducible tridiagonal. We call such a pair a Leonard pair on $V$. We introduce two canonical forms for Leonard pairs. We call these the TD-D canonical form and the LB-UB canonical form. In the TD-D canonical form the Leonard pair is represented by an irreducible tridiagonal matrix and a diagonal matrix, subject to a certain normalization. In the LB-UB canonical form the Leonard pair is represented by a lower bidiagonal matrix and an upper bidiagonal matrix, subject to a certain normalization. We describe the two canonical forms in detail. As an application we obtain the following results. Given square matrices $A,B$ over $\K$, with $A$ tridiagonal and $B$ diagonal, we display a necessary and sufficient condition for $A,B$ to represent a Leonard pair. Given square matrices $A,B$ over $\K$, with $A$ lower bidiagonal and $B$ upper bidiagonal, we display a necessary and sufficient condition for $A,B$ to represent a Leonard pair. We briefly discuss how Leonard pairs correspond to the $q$-Racah polynomials and some related polynomials in the Askey scheme. We present some open problems concerning Leonard pairs. | |
| dc.identifier | https://arxiv.org/abs/math/0304077 | |
| dc.identifier | http://arxiv.org/abs/math/0304077 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/66999 | |
| dc.subject | Rings and Algebras | |
| dc.subject | Quantum Algebra | |
| dc.subject | 05E35 | |
| dc.title | Two linear transformations each tri-diagonal with respect to an eigenbasis of the other; the TD-D canonical form and the LB-UB canonical form | |
| dc.type | text |