Let $V$ denote a vector space with finite positive dimension. We consider an ordered pair of linear transformations $A: V\rightarrow V$ and $A^{ *}: V\rightarrow V$ that satisfy (i) and (ii) below. (i) There exists a basis for $V$ with respect to which the matrix representing $A$ is irreducible tridiagonal and the matrix representing $A^{ *}$ is diagonal. (ii) There exists a basis for $V$ with respect to which the matrix representing $A^{ *}$ is irreducible tridiagonal and the matrix representing $A$ is diagonal. We call such a pair a Leonard pair on $V$. In the literature, there are some parameters that are used to describe Leonard pairs called the intersection numbers $\{ a_{ i}\}_{ i=0}^{ d}$, $\{ b_{ i}\}_{ i=0}^{ d-1}$, $\{ c_{ i}\}_{ i=1}^{ d}$, and the dual eigenvalues $\{ \theta^{ *}_{ i}\}_{ i=0}^{ d}$. In this paper, we provide two characterizations of Leonard pairs. For the first characterization, the focus is on the $\{ a_{ i}\}_{ i=0}^{ d}$ and $\{ \theta^{ *}_{ i}\}_{ i=0}^{ d}$. For the second characterization, the focus is on the $\{ b_{ i}\}_{ i=0}^{ d-1}$, $\{ c_{ i}\}_{ i=1}^{ d}$, and $\{ \theta^{ *}_{ i}\}_{ i=0}^{ d}$.
Categories: math.RA