Affine transformations of a Leonard pair
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Abstract
Let K denote a field and let V denote a vector space over K withfinite 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 withrespect to whichthe matrix representing A is irreducible
tridiagonal and the matrix representing A∗ is diagonal.
(ii) There exists a basis for V withrespect to whichthe matrix representing A∗ is irreducible
tridiagonal and the matrix representing A is diagonal.
Sucha pair is called a Leonard pair on V . Let ξ, ζ, ξ∗, ζ∗ denote scalars in K with ξ, ξ∗ nonzero, and
note that ξA + ζI, ξ∗A∗ + ζ∗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.