Two linear transformations each tridiagonal with respect to an eigenbasis of the other: comments on the split decomposition

JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS

卷:178

Article; Proceedings Paper

Terwilliger, P

关键词: Leonard pair;    tridiagonal pair;    q-Racah polynomial;   

DOI  :  10.1016/j.cam.2004.04.017

来源: Elsevier

【 摘 要 】

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 -> V and A* : V -> V that satisfy both conditions below: (i) There exists a basis for V with respect to which the matrix representing A is irreducible tridiagonal and the matrix representing A* is diagonal. ( ii) There exists a basis for V with respect to which the matrix representing A* is irreducible tridiagonal and the matrix representing A is diagonal. We call such a pair a Leonardpair on V. Referring to the above Leonard pair, it is known there exists a decomposition of V into a direct sum of one-dimensional subspaces, on which A acts in a lower bidiagonal fashion and A* acts in an upper bidiagonal fashion. This is called the split decomposition. In this paper, we give two characterizations of a Leonard pair that involve the split decomposition. (c) 2004 Elsevier B.V. All rights reserved.

【 授权许可】

Free