Affine transformations of a Leonard pair

Abstract

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 $A^{\ast }:V\to V$ 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 $A^{\ast }$ is diagonal. (ii) There exists a basis for $V$ with respect to which the matrix representing $A^{\ast }$ is irreducible tridiagonal and the matrix representing $A$ is diagonal. We call such a pair a Leonard pair on $V$. Let $x$, $c$, $x^{\ast }$, $c^{\ast }$ denote scalars in $K$ with $x$, $x^{\ast }$ nonzero, and note that $xA+cI$, $x^{\ast }{A}^{\ast }+c^{\ast }I$ is a Leonard pair on $V$. We give necessary and sufficient conditions for this Leonard pair to be isomorphic to the Leonard pair $A$, $A^{\ast }$. We also give necessary and sufficient conditions for this Leonard pair to be isomorphic to the Leonard pair $A^{\ast }$, $A$.

33 pages

Cited by in corpus (1)

• On the shape of a tridiagonal pair