The Enumeration of Jordan′s Normal Form of k-nilpotent Matrices

CHEN Jian-da; WANG Ping; ZHANG Lu

doi:10.15938/j.jhust.2019.04.023

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The Enumeration of Jordan′s Normal Form of k-nilpotent Matrices

更新时间：2026-08-31

- The Enumeration of Jordan′s Normal Form of k-nilpotent Matrices
- Vol. 24, Issue 4, Pages: 139-142(2019)
- 作者机构：
  
  1. 哈尔滨理工大学应用数学系,黑龙江,哈尔滨,150080
  2. 西安电子科技大学信息与通信工程,陕西,西安,710071
- 作者简介：
- 基金信息：
- DOI：10.15938/j.jhust.2019.04.023
  CLC： O151.21
- Published：2019
- 稿件说明：
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CHEN Jian-da, WANG Ping, ZHANG Lu. The Enumeration of Jordan′s Normal Form of k-nilpotent Matrices[J]. 2019, 24(4): 139-142.
DOI：

CHEN Jian-da, WANG Ping, ZHANG Lu. The Enumeration of Jordan′s Normal Form of k-nilpotent Matrices[J]. 2019, 24(4): 139-142. DOI： 10.15938/j.jhust.2019.04.023.

摘要

幂零矩阵作为一种特殊矩阵，具有很好的性质，这些性质使得幂零矩阵在密码学、构造认证码及安全性等方面应用非常广泛。首先针对n阶4-幂零矩阵的Jordan标准形的计数问题进行研究，通过对整数n的有序拆分方法，得到n阶4-幂零矩阵Jordan标准形的计数公式；其次讨论了当秩给定时的所有4-幂零矩阵，给出它的Jordan标准形的计数公式。最后针对k-幂零矩阵Jordan标准形的计数问题进行了研究，通过迭代的方法给出了n阶k-幂零矩阵的Jordan标准形的计数公式。

Abstract

As a kind of special matrices

the nilpotent matrices has good properties. In this paper

we mainly discuss the enumeration problem of the Jordan′s normal form of 4-nilpotent matrix. We get conclusions including the calculation formulas of the Jordan′s normal form of 4-nilpotent matrix and calculation formulas of the Jordan′s normal form of 4-nilpotent matrix with constant is presented. Then

We have the calculation formulas of Jordan′s normal form of n×n k-nilpotent matrix by iteration.

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