A pr 2 00 3 Two linear transformations each tri - diagonal with respect to an eigenbasis of the other ; the TD - D canonical form and the LB - UB canonical form ∗

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 B : V → V 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 K, 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 K, 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.