  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 $A:V\to V$ and $B:V\to V$ 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.
