循环 Hessenberg 对的变体
Variations on a circular Hessenberg pair
AI总结：
本文研究循环、拟循环、三对角和不可约三对角等类型的 Hessenberg 矩阵相关的 Hessenberg 系统族，目标是揭示这些族之间的关系，并依据参数数组描述每个族。
AI中文摘要：
当方阵次对角线以下的每个元素都为零且次对角线上的每个元素都非零时，该方阵称为 Hessenberg 矩阵。Hessenberg 对是在非零有限维向量空间上的一对可对角化线性映射，它们以 Hessenberg 方式作用于彼此的特征基。Hessenberg 系统 $\Phi$ 是 Hessenberg 对的一种‘定向’形式。已知 $\Phi$ 由其参数数组确定到同构，该参数数组由 $\Phi$ 的特征值序列、对偶特征值序列以及称为 $\Phi$ 的分裂序列的非零标量序列 $\{\phi_i\}_{i=1}^d$ 组成。我们对某些类型的 Hessenberg 矩阵感兴趣，它们被称为循环、拟循环、三对角和不可约三对角矩阵。我们关注其相关 Hessenberg 矩阵具有上述类型之一的 Hessenberg 系统族。不可约三对角类型的 Hessenberg 系统通常称为 Leonard 系统。在这种情况下，相关的 Hessenberg 对满足两个关系，称为三对角关系。我们关注其相关 Hessenberg 对满足三对角关系的 Hessenberg 系统族。我们还关注特征值序列和对偶特征值序列满足线性三项递推关系的 Hessenberg 系统族。在本文中，我们有两个主要目标。首先，我们展示上述 Hessenberg 系统族彼此之间是如何关联的。其次，我们根据参数数组描述每个族。
英文摘要：
A square matrix is called Hessenberg whenever each entry below the subdiagonal is zero and each entry on the subdiagonal is nonzero. A Hessenberg pair is an ordered pair of diagonalizable linear maps on a nonzero finite-dimensional vector space, that each act on an eigenbasis of the other one in a Hessenberg fashion. A Hessenberg system $Φ$ an `oriented' version of a Hessenberg pair. It is known that $Φ$ is determined up to isomorphism by its parameter array; this consists of the eigenvalue sequence of $Φ$, the dual eigenvalue sequence of $Φ$, and a sequence of nonzero scalars $\{ϕ_i\}_{i=1}^d$ called the split sequence of $Φ$. We are interested in some types of Hessenberg matrices, said to be circular, quasi-circular, tridiagonal, and irreducible tridiagonal. We are interested in the families of Hessenberg systems for which the associated Hessenberg matrices have one of the above types. A Hessenberg system of irreducible tridiagonal type is often called a Leonard system. In this case the associated Hessenberg pair satisfies two relations, called the tridiagonal relations. We are interested in the family of Hessenberg systems for which the associated Hessenberg pair satisfies the tridiagonal relations. We are also interested in the family of Hessenberg systems for which the eigenvalue sequence and dual eigenvalue sequence satisfy a linear three-term recurrence. In the present paper we have two main goals. First, we show how the above families of Hessenberg systems are related to each other. Second, we describe each family in terms of the parameter array.
