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Abstract <p>We study fields over which every matrix can be represented as a sum of two potent matrices and a nilpotent matrix. In particular, it is shown that over a field <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(P\)</EquationSource> <!--RusMath2570079Abyzov-m1--> </InlineEquation> every matrix can be represented as a sum of two idempotent matrices and a nilpotent matrix exactly when either <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(P \cong {{\mathbb{F}}_{2}},\)</EquationSource> <!--RusMath2570079Abyzov-m2--> </InlineEquation> or <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(P \cong {{\mathbb{F}}_{3}}\)</EquationSource> <!--RusMath2570079Abyzov-m3--> </InlineEquation>.</p>

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Fields over Which Matrices Can Be Represented As the Sum of Potent and Nilpotent Matrices

  • 期刊论文

  • 发表日期：

    2025年10月1日

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  • A. N. Abyzov,
  • D. T. Tapkin

摘要

Abstract

We study fields over which every matrix can be represented as a sum of two potent matrices and a nilpotent matrix. In particular, it is shown that over a field \(P\) every matrix can be represented as a sum of two idempotent matrices and a nilpotent matrix exactly when either \(P \cong {{\mathbb{F}}_{2}},\) or \(P \cong {{\mathbb{F}}_{3}}\) .