Two Linear Transformations Each Tri-Diagonal with Respect to an Eigenbasis of the Other; the - Canonical Form and the - Canonical Form

Let denote a field and let 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 with respect to which the matrix representing is irreducible tridiagonal and the matrix representing is diagonal.(ii) There exists a basis for with respect to which the matrix representing is diagonal and the matrix representing is irreducible tridiagonal.We call such a pair a on . We introduce two canonical forms for Leonard pairs. We call these the and the . In the - canonical form the Leonard pair is represented by an irreducible tridiagonal matrix and a diagonal matrix, subject to a certain normalization. In the - 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 over K, with tridiagonal and diagonal, we display a necessary and su cient condition for to represent a Leonard pair. Given square matrices over K, with lower bidiagonal and upper bidiagonal, we display a necessary and su cient condition for to represent a Leonard pair. We briefly discuss how Leonard pairs correspond to the -Racah polynomials and some related polynomials in the Askey scheme. We present some open problems concerning Leonard pairs