Circular bidiagonal pairs

Abstract

A square matrix is said to be circular bidiagonal whenever (i) each nonzero entry is on the diagonal, or the subdiagonal, or in the top-right corner; (ii) each subdiagonal entry is nonzero, and the entry in the top-right corner is nonzero. Let $\mathbb{F}$ denote a field, and let $V$ denote a nonzero finite-dimensional vector space over $\mathbb{F}$. We consider an ordered pair of $\mathbb{F}$-linear maps $A:V\to V$ and $A^{\ast }:V\to V$ that satisfy the following two conditions: (i) there exists a basis for $V$ with respect to which the matrix representing $A$ is circular bidiagonal and the matrix representing $A^{\ast }$ is diagonal; (ii) there exists a basis for $V$ with respect to which the matrix representing $A^{\ast }$ is circular bidiagonal and the matrix representing $A$ is diagonal. We call such a pair a circular bidiagonal pair on $V$. We classify the circular bidiagonal pairs up to affine equivalence. There are two infinite families of solutions, which we describe in detail.

30 pages

References in corpus (1)

• Circular Hessenberg Pairs