Let K denote a field, and let V denote a vector space over K with finite positive dimension. Consider a pair of linear transformations A: V -> V and B: V -> V that satisfy (i), (ii) below:

(i) there exists a basis for V with respect to which the matrix representing A is diagonal and the matrix representing B is irreducible tridiagonal,

(ii) there exists a basis for V with respect to which the matrix representing B is diagonal and the matrix representing A is irreducible tridiagonal. (A tridiagonal matrix is said to be irreducible whenever the entries immediately above and below the main diagonal are nonzero).

We call such a pair A,B a "Leonard pair". (The name comes from a connection to a theorem of Leonard involving the q-Racah polynomials).

Leonard pairs are interesting on several levels. First of all, they are elegant linear algebraic objects and there is a very nice classification. Secondly, Leonard pairs are associated with a family of orthogonal polynomials, consisting of the q-Racah polynomials and their relatives. The q-Racah polynomials describe the Racah coefficients for representations of the quantum algebra U_q(sl_2), and also play a role in the quantum theory of angular momentum. Roughly speaking, it is appropriate to think of Leonard pairs as "q-Racah polynomials in disguise". Thirdly, Leonard pairs all "come from" irreducible representations of a certain algebra, called the Tridiagonal algebra. This algebra is a kind of generalization of U_q(sl_2). Fourthly, Leonard pairs arise naturally in combinatorics, in the context of distance-regular graphs. A distance-regular graph is a finite, undirected, connected graph that is "highly regular" in a certain way. These distance-regular graphs support representations of the Tridiagonal algebra, and these representations give Leonard pairs.