循环 Hessenberg 对
Circular Hessenberg Pairs
AI总结：
本文研究循环 Hessenberg 对，构造六个基及其转移矩阵和表示矩阵，引入循环型循环 Hessenberg 对，证明其与三对角关系的等价性，并通过四个系统族完成循环循环 Hessenberg 系统的同构分类。
AI中文摘要：
一个方阵若其次对角线以下的每个元素均为零且次对角线上的每个元素均非零，则称为 Hessenberg 矩阵。设 $M$ 表示一个 Hessenberg 矩阵。若 $M$ 的右上角元素非零且超对角线以上的所有其他元素均为零，则称 $M$ 为循环的。一个循环 Hessenberg 对由非零有限维向量空间上的两个可对角化线性映射组成，其中每个映射都以循环 Hessenberg 方式作用于另一个映射的特征基。设 $A, A^*$ 表示一个循环 Hessenberg 对。我们研究了底向量空间的六个我们认为有吸引力的基。我们展示了这六个基中某些基对之间的转移矩阵。我们还展示了 $A$ 和 $A^*$ 关于这六个基的表示矩阵。我们引入了一类特殊的循环 Hessenberg 对，称为循环的。我们证明循环 Hessenberg 对 $A, A^*$ 是循环的当且仅当 $A, A^*$ 满足三对角关系。对于循环 Hessenberg 对，存在一个相关对象，称为循环 Hessenberg 系统。我们在同构意义下对循环的循环 Hessenberg 系统进行分类。为此，我们构造了四个循环的循环 Hessenberg 系统族。我们证明每个循环的循环 Hessenberg 系统都同构于这四个族之一的成员。
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. Let $M$ denote a Hessenberg matrix. Then $M$ is called circular whenever the upper-right corner entry of $M$ is nonzero and every other entry above the superdiagonal is zero. A circular Hessenberg pair consists of two diagonalizable linear maps on a nonzero finite-dimensional vector space, that each act on an eigenbasis of the other one in a circular Hessenberg fashion. Let $A, A^*$ denote a circular Hessenberg pair. We investigate six bases for the underlying vector space that we find attractive. We display the transition matrices between certain pairs of bases among the six. We also display the matrices that represent $A$ and $A^*$ with respect to the six bases. We introduce a special type of circular Hessenberg pair, said to be recurrent. We show that a circular Hessenberg pair $A, A^*$ is recurrent if and only if $A, A^*$ satisfy the tridiagonal relations. For a circular Hessenberg pair, there is a related object called a circular Hessenberg system. We classify up to isomorphism the recurrent circular Hessenberg systems. To this end, we construct four families of recurrent circular Hessenberg systems. We show that every recurrent circular Hessenberg system is isomorphic to a member of one of the four families.
