ELA, Volume 16, pp. 389-418, December 2007, abstract.

Affine Transformations of a Leonard Pair

Kazumasa Nomura and Paul Terwilliger

Let K denote a field and let V denote a vector space over K with finite 
positive dimension. An ordered pair is considered of linear transformations 
A : V --> V and A^* : V --> 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.

Such a pair is called a {\em Leonard pair} on V. Let x, z, x^*, z^* 
denote scalars in K with x, x^* nonzero, and note that xA+zI, x^*A^*+z^*I 
is a Leonard pair on V. Necessary and sufficient conditions are given for 
this Leonard pair to be isomorphic to A, A^*. Also given are necessary and 
sufficient conditions for this Leonard pair to be isomorphic to the
Leonard pair A^*, A.