Two linear transformations each tridiagonal with respect to an eigenbasis of the other; comments on the parameter array

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 conditions (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 $A^{\ast }$ is diagonal. (ii) There exists a basis for $V$ with respect to which the matrix representing $A$ is diagonal and the matrix representing $A^{\ast }$ is irreducible tridiagonal. We call such a pair a {\it Leonard pair} on $V$. The structure of any given Leonard pair is deterined by a certain sequence of scalars called its {\it parameter array}. The set of parameter arrays is an affine algebraic variety. We give two characterizations of this variety. One involves bidiagonal matrices and the other involves orthogonal polynomials.

26 pages

References in corpus (1)

• Two linear transformations each tridiagonal with respect to an eigenbasis of the other; comments on the split decomposition

Cited by in corpus (8)

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