Let F denote a field, and let V denote a vector space over F with finite positive dimension. We consider a pair of linear transformations A : V → V and A * : V → V satisfying 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. Integral to our proof is the following result. We show that for any Leonard pair A, A * on V , there exists a sequence of scalars β, γ, γ * , ̺, ̺ * taken from F such that bothwhere [r, s] means rs − sr. The sequence is uniquely determined by the Leonard pair if the dimension of V is at least 4. We conclude by showing how Leonard systems correspond to q-Racah and related polynomials from the Askey scheme.

Inspired by the theory of P -and Q-polynomial association schemes we consider the following situation in linear algebra. Let F denote a field, and let V denote a vector space over F with finite positive dimension. We consider a pair of linear transformations A : V → V and A * : V → V satisfying the following four conditions. θ * i−1 − θ * i both equal β + 1, for 2 ≤ i ≤ d − 1. We hope these results will ultimately lead to a complete classification of the TD pairs.

We define an algebra on two generators which we call the Tridiagonal algebra, and we consider its irreducible modules. The algebra is defined as follows. Let K denote a field, and let β, γ, γ * , ̺, ̺ * denote a sequence of scalars taken from K. The corresponding Tridiagonal algebra T is the associative K-algebra with 1 generated by two symbols A, A * subject to the relations [

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 which satisfy the following two properties:(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 Leonard pair on V . Referring to the above Leonard pair, we show there exists a sequence of scalars β, γ, γ * , ̺, ̺ * , ω, η, η * taken from K such that bothThe sequence is uniquely determined by the Leonard pair provided the dimension of V is at least 4. The equations above are called the Askey-Wilson relations.