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Abstract <p> Let <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11006_2025_3080_Article_IEq1.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="40" /> </InlineMediaObject> <EquationSource Format="TEX">\(\mathbb{R}^{m \times n}\)</EquationSource> </InlineEquation> be the vector space of <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11006_2025_3080_Article_IEq2.gif" Format="GIF" Height="13" Rendition="HTML" Resolution="72" Type="Linedraw" Width="49" /> </InlineMediaObject> <EquationSource Format="TEX">\(m \times n\)</EquationSource> </InlineEquation> real matrices, and let <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11006_2025_3080_Article_IEq3.gif" Format="GIF" Height="18" Rendition="HTML" Resolution="72" Type="Linedraw" Width="137" /> </InlineMediaObject> <EquationSource Format="TEX">\(\phi\colon \mathbb{R}^{m \times n} \longrightarrow \mathbb{R}^{m \times n}\)</EquationSource> </InlineEquation> be a linear transformation such that <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11006_2025_3080_Article_IEq4.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="136" /> </InlineMediaObject> <EquationSource Format="TEX">\(\operatorname{vol}(\phi(A)) = \operatorname{vol}(A)\)</EquationSource> </InlineEquation> for all <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11006_2025_3080_Article_IEq5.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="74" /> </InlineMediaObject> <EquationSource Format="TEX">\(A \in \mathbb{R}^{m \times n}\)</EquationSource> </InlineEquation>. If <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11006_2025_3080_Article_IEq6.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="50" /> </InlineMediaObject> <EquationSource Format="TEX">\(m \neq n\)</EquationSource> </InlineEquation>, then there exist two orthogonal matrices <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11006_2025_3080_Article_IEq7.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="77" /> </InlineMediaObject> <EquationSource Format="TEX">\(P \in \mathbb{R}^{m \times m}\)</EquationSource> </InlineEquation> and <InlineEquation ID="IEq8"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11006_2025_3080_Article_IEq8.gif" Format="GIF" Height="18" Rendition="HTML" Resolution="72" Type="Linedraw" Width="72" /> </InlineMediaObject> <EquationSource Format="TEX">\(Q \in \mathbb{R}^{n \times n}\)</EquationSource> </InlineEquation> such that <InlineEquation ID="IEq9"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11006_2025_3080_Article_IEq9.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="102" /> </InlineMediaObject> <EquationSource Format="TEX">\(\phi(A) = P A Q\)</EquationSource> </InlineEquation> for all <InlineEquation ID="IEq10"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11006_2025_3080_Article_IEq5.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="74" /> </InlineMediaObject> <EquationSource Format="TEX">\(A \in \mathbb{R}^{m \times n}\)</EquationSource> </InlineEquation>. If <InlineEquation ID="IEq11"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11006_2025_3080_Article_IEq11.gif" Format="GIF" Height="10" Rendition="HTML" Resolution="72" Type="Linedraw" Width="50" /> </InlineMediaObject> <EquationSource Format="TEX">\(m = n\)</EquationSource> </InlineEquation>, then there exist two orthogonal matrices <InlineEquation ID="IEq12"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11006_2025_3080_Article_IEq12.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="71" /> </InlineMediaObject> <EquationSource Format="TEX">\(P \in \mathbb{R}^{n \times n}\)</EquationSource> </InlineEquation> and <InlineEquation ID="IEq13"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11006_2025_3080_Article_IEq8.gif" Format="GIF" Height="18" Rendition="HTML" Resolution="72" Type="Linedraw" Width="72" /> </InlineMediaObject> <EquationSource Format="TEX">\(Q \in \mathbb{R}^{n \times n}\)</EquationSource> </InlineEquation> such that either <InlineEquation ID="IEq14"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11006_2025_3080_Article_IEq9.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="102" /> </InlineMediaObject> <EquationSource Format="TEX">\(\phi(A) = P A Q\)</EquationSource> </InlineEquation> for all <InlineEquation ID="IEq15"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11006_2025_3080_Article_IEq15.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="70" /> </InlineMediaObject> <EquationSource Format="TEX">\(A \in \mathbb{R}^{n \times n}\)</EquationSource> </InlineEquation> or <InlineEquation ID="IEq16"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11006_2025_3080_Article_IEq16.gif" Format="GIF" Height="21" Rendition="HTML" Resolution="72" Type="Linedraw" Width="111" /> </InlineMediaObject> <EquationSource Format="TEX">\(\phi(A) = P A^{\mathrm T} Q\)</EquationSource> </InlineEquation> for all <InlineEquation ID="IEq17"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11006_2025_3080_Article_IEq15.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="70" /> </InlineMediaObject> <EquationSource Format="TEX">\(A \in \mathbb{R}^{n \times n}\)</EquationSource> </InlineEquation>. </p>

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A Note on Matrix Volume Preservers

  • 期刊论文

  • 发表日期：

    2025年6月1日

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  • Z. Zhou

摘要

Abstract

Let \(\mathbb{R}^{m \times n}\) be the vector space of \(m \times n\) real matrices, and let \(\phi\colon \mathbb{R}^{m \times n} \longrightarrow \mathbb{R}^{m \times n}\) be a linear transformation such that \(\operatorname{vol}(\phi(A)) = \operatorname{vol}(A)\) for all \(A \in \mathbb{R}^{m \times n}\) . If \(m \neq n\) , then there exist two orthogonal matrices \(P \in \mathbb{R}^{m \times m}\) and \(Q \in \mathbb{R}^{n \times n}\) such that \(\phi(A) = P A Q\) for all \(A \in \mathbb{R}^{m \times n}\) . If \(m = n\) , then there exist two orthogonal matrices \(P \in \mathbb{R}^{n \times n}\) and \(Q \in \mathbb{R}^{n \times n}\) such that either \(\phi(A) = P A Q\) for all \(A \in \mathbb{R}^{n \times n}\) or \(\phi(A) = P A^{\mathrm T} Q\) for all \(A \in \mathbb{R}^{n \times n}\) .