<p>In this article, we introduce and study properties of the Thompson metric induced by the chaotic order on the cone of positive definite matrices. Furthermore, we introduce an approach to characterize geodesic curves associated to Thompson metrics and study in-betweenness type results for the metric induced by the chaotic order. Among other results, we obtain that the map <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11117_2025_1119_Article_IEq1.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="239" /> </InlineMediaObject> <EquationSource Format="TEX">$A\mapsto d_\ll (A,Q_{\alpha ,z}(B,A)); \quad z>0,$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>A</mi> <mo>↦</mo> <msub> <mi>d</mi> <mo>≪</mo> </msub> <mrow> <mo stretchy="false">(</mo> <mi>A</mi> <mo>,</mo> <msub> <mi>Q</mi> <mrow> <mi>α</mi> <mo>,</mo> <mi>z</mi> </mrow> </msub> <mrow> <mo stretchy="false">(</mo> <mi>B</mi> <mo>,</mo> <mi>A</mi> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">)</mo> </mrow> <mo>;</mo> <mspace width="1em" /> <mi>z</mi> <mo>></mo> <mn>0</mn> <mo>,</mo> </mrow> </math></EquationSource> </InlineEquation> satisfies Audenaert’s in-betweenness for <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11117_2025_1119_Article_IEq2.gif" Format="GIF" Height="15" Rendition="HTML" Resolution="72" Type="Linedraw" Width="74" /> </InlineMediaObject> <EquationSource Format="TEX">$0\le \alpha \le 1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>0</mn> <mo>≤</mo> <mi>α</mi> <mo>≤</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>.</p>

On the chaotic Thompson metric of matrices

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发表日期：

2025年3月29日

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Małgorzata M. Czerwińska,
Raluca Dumitru,
Jose A. Franco

摘要

In this article, we introduce and study properties of the Thompson metric induced by the chaotic order on the cone of positive definite matrices. Furthermore, we introduce an approach to characterize geodesic curves associated to Thompson metrics and study in-betweenness type results for the metric induced by the chaotic order. Among other results, we obtain that the map $A\mapsto d_\ll (A,Q_{\alpha ,z}(B,A)); \quad z>0,$ $A \mapsto d_{≪} (A, Q_{α, z} (B, A)); z > 0,$ satisfies Audenaert’s in-betweenness for $0\le \alpha \le 1$ $0 \leq α \leq 1$ .