In this survey paper we give an elementary introduction to the theory of Leonard pairs. A Leonard pair is
de.ned as follows. Let K denote a .eld and let V denote a vector space over K with .nite positive dimension.
By a Leonard pair on V we mean an ordered pair of linear transformations A : V → V and B : V → V that
satisfy conditions (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 give several examples of Leonard pairs. Using these we illustrate how Leonard pairs arise in representation
theory, combinatorics, and the theory of orthogonal polynomials.