Two linear transformations each tridiagonal with respect to an eigenbasis of the other

Details Two linear transformations each tridiagonal with respect to an eigenbasis of the other Two linear transformations each tridiagonal with respect to an eigenbasis of the other

2004-06-27

arxiv.org/abs/math/0406555v1

Linear Algebra Appl. 330 (2001), 149--203

Mathematics

Rings and Algebras

Scientific paper

Let $K$ denote a field and let $V$ denote a vector space over $K$ with finite positive dimension. We consider a pair of linear transformations $A:V\to V$ and $A^*:V\to V$ that satisfy both conditions below: (i) There exists a basis for $V$ with respect to which the matrix representing $A$ is diagonal, and the matrix representing $A^*$ is irreducible tridiagonal. (ii) There exists a basis for $V$ with respect to which the matrix representing $A^*$ is diagonal, and the matrix representing $A$ is irreducible tridiagonal. We call such a pair a Leonard pair on $V$. Refining this notion a bit, we introduce the concept of a Leonard system. We give a complete classification of Leonard systems. We discuss how Leonard systems correspond to the $q$-Racah and related polynomials from the Askey scheme.

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