ul 2 00 3 Two linear transformations each tridiagonal with respect to an eigenbasis of the other ; an overview

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 conditions (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 A∗ is diagonal. (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 . We give an overview of the theory of Leonard pairs. 1 Leonard pairs We begin by recalling the notion of a Leonard pair. We will use the following terms. Let X denote a square matrix. Then X is called tridiagonal whenever each nonzero entry lies on either the diagonal, the subdiagonal, or the superdiagonal. Assume X is tridiagonal. Then X is called irreducible whenever each entry on the subdiagonal is nonzero and each entry on the superdiagonal is nonzero. We now define a Leonard pair. For the rest of this paper K will denote a field. Definition 1.1 [37] Let V denote a vector space over K with finite positive dimension. By a Leonard pair on V we mean an ordered pair of linear transformations A : V → V and A∗ : V → V that satisfy conditions (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 A∗ is diagonal. (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.