不可约上Hessenberg矩阵的交换子空间的一些讨论

Discussion on the Commutative Subspace of Unreduced Upper Hessenberg Matrices

得到了n阶不可约上Hessenberg矩阵A的交换子空间以E,A,…,A^(n-1)为基的结论,同时给出了在数学专业本科高等代数知识平台上的证明方法。这样,2015年硕士研究生入学考试的高等代数试卷的相应题目所要求的"A有n个线性无关的特征向量"是可去掉的。

It was proved that, E, A, …, A^n-1 is a basis of the commutative subspace of an n by n unreduced upper Hessenberg matrix A. Meanwhile, the proof over the platform for undergraduate course of Advance Algebra of mathematics specialty was shown. Therefore, in some related text questions in the test papers of Advanced Algebra of 2015 postgraduate entrance examinations, the condition ＂A has n linear independent vectors＂ is unnecessary.