Two linear transformations each tridiagonal with respect to an eigenbasis of the other; the TD–D canonical form and the LB–UB canonical form

Let denote a field and let V denote a vector space over with finite positive dimension. We consider an ordered pair of linear transformations and 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 , 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 , 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.