论文标题
圆形黑森贝格对
Circular Hessenberg Pairs
论文作者
论文摘要
每当下方下方的每个条目为零，并且子数字上的每个条目均非零时，平方矩阵就称为Hessenberg。令$ m $表示Hessenberg Matrix。然后，每当$ m $的上右上角入口为nonzero时，$ m $被称为圆形，超过级别的其他条目为零。一对圆形黑森贝格对由两个有限二维矢量空间上的两个可对角线线性图组成，每个图都以圆形的hessenberg时尚的方式作用于另一个的特征性。令$ a，a^*$表示圆形的黑森贝格对。我们研究了我们发现有吸引力的基础向量空间的六个基础。我们显示了六个基碱基之间的过渡矩阵。我们还显示代表$ a $和$ a^*$的矩阵。我们引入了一种特殊类型的圆形黑森贝格对，据说是经常出现的。我们表明，当且仅在$ a时，a^*$ a^*$的圆形Hessenberg对$ a，a^*$会经常出现。对于圆形的黑森伯格对，有一个称为圆形黑森贝格系统的相关对象。我们将复发性圆形黑森贝格系统分类为同构。为此，我们构建了四个经常性的圆形赫森伯格系统家族。我们表明，每个经常性的圆形赫森伯格系统对四个家庭之一的成员都是同构的。
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.
