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Abstract <p>Let <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11968_2025_5146_Article_IEq1.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="17" /> </InlineMediaObject> <EquationSource Format="TEX">\(A\)</EquationSource> <!--CMatCMGU2470032Ikramov-m1--> </InlineEquation> and <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11968_2025_5146_Article_IEq2.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="17" /> </InlineMediaObject> <EquationSource Format="TEX">\(B\)</EquationSource> <!--CMatCMGU2470032Ikramov-m2--> </InlineEquation> be matrices of order <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11968_2025_5146_Article_IEq3.gif" Format="GIF" Height="10" Rendition="HTML" Resolution="72" Type="Linedraw" Width="14" /> </InlineMediaObject> <EquationSource Format="TEX">\(n\)</EquationSource> <!--CMatCMGU2470032Ikramov-m3--> </InlineEquation> that are direct sums of nilpotent Jordan blocks. Suppose that <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11968_2025_5146_Article_IEq4.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="17" /> </InlineMediaObject> <EquationSource Format="TEX">\(A\)</EquationSource> <!--CMatCMGU2470032Ikramov-m4--> </InlineEquation> and <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11968_2025_5146_Article_IEq5.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="17" /> </InlineMediaObject> <EquationSource Format="TEX">\(B\)</EquationSource> <!--CMatCMGU2470032Ikramov-m5--> </InlineEquation> are not just different arrangements of the same blocks but, rather, they differ in the sizes of the blocks. It is shown that, in this case, <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11968_2025_5146_Article_IEq6.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="17" /> </InlineMediaObject> <EquationSource Format="TEX">\(A\)</EquationSource> <!--CMatCMGU2470032Ikramov-m6--> </InlineEquation> and <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11968_2025_5146_Article_IEq7.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="17" /> </InlineMediaObject> <EquationSource Format="TEX">\(B\)</EquationSource> <!--CMatCMGU2470032Ikramov-m7--> </InlineEquation> cannot be congruent. This result can be regarded as a new proof of the uniqueness of the singular part in the Horn–Sergeichuk canonical form of a singular matrix.</p>

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On the Noncongruence of Direct Sums of Nilpotent Jordan Blocks

  • 期刊论文

  • 发表日期：

    2025年3月29日

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  • Kh. D. Ikramov

摘要

Abstract

Let \(A\) and \(B\) be matrices of order \(n\) that are direct sums of nilpotent Jordan blocks. Suppose that \(A\) and \(B\) are not just different arrangements of the same blocks but, rather, they differ in the sizes of the blocks. It is shown that, in this case, \(A\) and \(B\) cannot be congruent. This result can be regarded as a new proof of the uniqueness of the singular part in the Horn–Sergeichuk canonical form of a singular matrix.