Article ID | Journal | Published Year | Pages | File Type |
---|---|---|---|---|
9498285 | Linear Algebra and its Applications | 2005 | 25 Pages |
Abstract
Let (A, A*) denote a tridiagonal pair on a vector space V over a field K. Let V0, â¦Â , Vd denote a standard ordering of the eigenspaces of A on V, and let θ0, â¦Â , θd denote the corresponding eigenvalues of A. We assume d ⩾ 3. Let q denote a scalar taken from the algebraic closure of K such that q2 + qâ2 + 1 = (θ3 â θ0)/(θ2 â θ1). We assume q is not a root of unity. Let Ïi denote the dimension of Vi. The sequence Ï0, Ï1, â¦Â , Ïd is called the shape of the tridiagonal pair. It is known there exists a unique integer h (0 ⩽ h ⩽d/2) such that Ïiâ1 < Ïi for 1 ⩽ i ⩽ h, Ïiâ1 = Ïi for h < i ⩽ d â h, and Ïiâ1 > Ïi for d â h < i ⩽ d. The integer h is known as the height of the tridiagonal pair. In this paper we show that the shape of a tridiagonal pair of height one with Ï0 = 1 is either 1, 2, 2, â¦Â , 2, 1 or 1, 3, 3, 1. In each case, we display a basis for V and give the action of A, A* on this basis.
Related Topics
Physical Sciences and Engineering
Mathematics
Algebra and Number Theory
Authors
Kazumasa Nomura,