The switching element for a Leonard pair

Details The switching element for a Leonard pair The switching element for a Leonard pair

2006-08-25

arxiv.org/abs/math/0608623v1

Mathematics

Rings and Algebras

29 pages

Scientific paper

Let $V$ denote a vector space with finite positive dimension. We consider a pair of linear transformations $A : V \to V$ and $A^* : V \to V$ 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 $A^*$ 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. We call such a pair a {\em Leonard pair} on $V$. Let $v_0,v_1,...,v_d$ (resp. $w_0,w_1,...,w_d$) denote a basis for $V$ referred to in (i) (resp. (ii)). We show that there exists a unique linear transformation $S: V \to V$ that sends $v_0$ to a scalar multiple of $v_d$, fixes $w_0$, and sends $w_i$ to a scalar multiple of $w_i$ for $1 \leq i \leq d$. We call $S$ the {\it switching element}. We describe $S$ from many points of view.

Affiliated with

Also associated with

No associations

Rating

The switching element for a Leonard pair does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with The switching element for a Leonard pair, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and The switching element for a Leonard pair will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-111867