Sums of two square-zero matrices over an arbitrary field

被引：19

作者：

Botha, J. D. ^{[1
]}

机构：

[1] Univ S Africa, Dept Math Sci, ZA-0003 Pretoria, South Africa

来源：

LINEAR ALGEBRA AND ITS APPLICATIONS | 2012年 / 436卷 / 03期

关键词：

Square-zero; Nilpotent; Companion matrix; Nonderogatory; Similarity; Rational canonical form; Trace; Field; Characteristic two; Algebraic closure;

D O I：

10.1016/j.laa.2011.07.011

中图分类号：

O29 [应用数学];

学科分类号：

070104 ;

摘要：

The problem to express an n x n matrix A as the sum of two square-zero matrices was first investigated by Wang and Wu [2] for matrices over the complex field. This paper investigates the problem over an arbitrary field F. It is shown that, if char(F) not equal 2, then A E M(n) (F) is the sum of two square-zero matrices if and only if A is similar to a matrix of the form N circle plus X circle plus (-X) circle plus (circle plus(m)(i=1) C(gi(x(2)))), where N is nilpotent, X is nonsingular, and each C(gi (x2)) is a companion matrix associated with an even-power polynomial with nonzero constant term. If F is of characteristic two, the term X circle plus (-X) falls away. If F is of characteristic zero and algebraically closed, the term circle plus(m)(i=1) C(gi (x(2))) falls away and the result of Wang and Wu is obtained. (C) 2011 Elsevier Inc. All rights reserved.

引用

页码：516 / 524

页数：9