Two linear transformations each tridiagonal with respect to an eigenbasis of the other; an overview

Details Two linear transformations each tridiagonal with respect to an eigenbasis of the other; an overview Two linear transformations each tridiagonal with respect to an eigenbasis of the other; an overview

2003-07-04

arxiv.org/abs/math/0307063v1

Mathematics

Rings and Algebras

14 pages, 1 figure

Scientific paper

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 $A^*:V\to V$ 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 $A^*$ is diagonal. (ii) There exists a basis for $V$ with respect to which the matrix representing $A$ is diagonal and the matrix representing $A^*$ is irreducible tridiagonal. We call such a pair a Leonard pair on $V$. We give an overview of the theory of Leonard pairs.

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