  Let $K$ denote a field, and let $V$ denote a vector space over $K$ with
finite positive dimension. Consider a pair of linear transformations $A:V\to V$
and $A^*:V\to V$ that satisfy both conditions below:
  (i) There exists a basis for $V$ with respect to which the matrix
representing $A$ is diagonal, and the matrix representing $A^*$ is irreducible
tridiagonal.
  (ii) There exists a basis for $V$ with respect to which the matrix
representing $A^*$ is diagonal, and the matrix representing $A$ is irreducible
tridiagonal.
  Such a pair is called a Leonard pair on $V$. In this paper we introduce a
mild generalization of a Leonard pair called a tridiagonal pair. A Leonard pair
is the same thing as a tridiagonal pair such that for each transformation all
eigenspaces have dimension one.
