Cross-positive matrices revisited
Identifieur interne : 002910 ( Main/Exploration ); précédent : 002909; suivant : 002911Cross-positive matrices revisited
Auteurs : Peter Gritzmann [Allemagne] ; Victor Klee [États-Unis, Allemagne] ; Bit-Shun Tam [République populaire de Chine, Allemagne]Source :
- Linear Algebra and Its Applications [ 0024-3795 ] ; 1995.
Abstract
For a closed, pointed n-dimensional convex cone K in Rn, let π(K) denote the set of all n × n real matrices A which as linear operators map K into itself. Let ∑(K) denote the set of all n × n matrices that are cross-positive on K, and L(K) = ∑(K) ∩ [− ∑(K)], the lineality space of ∑(K). Let Λ = RI, the set of all real multiples of the n × n identity matrix I. Then π(K)+Δ⊆π(K)+L(K)⊆cl[π(K)+Δ]=Σ(K). The final equality was proved in 1970 by Schneider and Vidyasagar, who showed also that π(K) + Λ = ∑(K) when K is polyhedral but not when K is a three-dimensional circular cone. They asked for a general characterization of those K for which the equality holds. It is shown here that if n ⩾ 3 and the cone K is strictly convex or smooth, then π(K) + Λ ≠ ∑(K); hence for n ⩾ 3 the equality fails for “almost all” K in the sense of Baire category. However, the equality does hold for some nonpolyhedral K, as was shown by a construction that appeared in the third author's 1977 dissertation and is explained here in more detail. In 1994 it was shown by Stern and Wolkowicz that the weaker equality π(K) + L(K) = ∑(K) holds for all ellipsoidal (as well as for all polyhedral) K, and they wondered whether this equality holds for all K. However, their equality certainly fails for all strictly convex or smooth K such that L(K) = Λ, and it is shown here that this also includes “almost all” K when n ⩾ 3.
Url:
DOI: 10.1016/0024-3795(93)00364-6
Affiliations:
Links toward previous steps (curation, corpus...)
- to stream Istex, to step Corpus: 001A37
- to stream Istex, to step Curation: 001920
- to stream Istex, to step Checkpoint: 001107
- to stream Main, to step Merge: 002E12
- to stream Main, to step Curation: 002910
Le document en format XML
<record><TEI wicri:istexFullTextTei="biblStruct"><teiHeader><fileDesc><titleStmt><title>Cross-positive matrices revisited</title>
<author><name sortKey="Gritzmann, Peter" sort="Gritzmann, Peter" uniqKey="Gritzmann P" first="Peter" last="Gritzmann">Peter Gritzmann</name>
</author>
<author><name sortKey="Klee, Victor" sort="Klee, Victor" uniqKey="Klee V" first="Victor" last="Klee">Victor Klee</name>
</author>
<author><name sortKey="Tam, Bit Shun" sort="Tam, Bit Shun" uniqKey="Tam B" first="Bit-Shun" last="Tam">Bit-Shun Tam</name>
</author>
</titleStmt>
<publicationStmt><idno type="wicri:source">ISTEX</idno>
<idno type="RBID">ISTEX:070F48080B858C2F6DCB61E9A4F48C7723DD1653</idno>
<date when="1995" year="1995">1995</date>
<idno type="doi">10.1016/0024-3795(93)00364-6</idno>
<idno type="url">https://api.istex.fr/document/070F48080B858C2F6DCB61E9A4F48C7723DD1653/fulltext/pdf</idno>
<idno type="wicri:Area/Istex/Corpus">001A37</idno>
<idno type="wicri:explorRef" wicri:stream="Istex" wicri:step="Corpus" wicri:corpus="ISTEX">001A37</idno>
<idno type="wicri:Area/Istex/Curation">001920</idno>
<idno type="wicri:Area/Istex/Checkpoint">001107</idno>
<idno type="wicri:explorRef" wicri:stream="Istex" wicri:step="Checkpoint">001107</idno>
<idno type="wicri:doubleKey">0024-3795:1995:Gritzmann P:cross:positive:matrices</idno>
<idno type="wicri:Area/Main/Merge">002E12</idno>
<idno type="wicri:Area/Main/Curation">002910</idno>
<idno type="wicri:Area/Main/Exploration">002910</idno>
</publicationStmt>
<sourceDesc><biblStruct><analytic><title level="a">Cross-positive matrices revisited</title>
<author><name sortKey="Gritzmann, Peter" sort="Gritzmann, Peter" uniqKey="Gritzmann P" first="Peter" last="Gritzmann">Peter Gritzmann</name>
<affiliation wicri:level="1"><country xml:lang="fr">Allemagne</country>
<wicri:regionArea>Universität Trier Fb IV, Mathematik D-54286 Trier</wicri:regionArea>
<wicri:noRegion>54286 Trier</wicri:noRegion>
<wicri:noRegion>Mathematik D-54286 Trier</wicri:noRegion>
</affiliation>
<affiliation wicri:level="1"><country wicri:rule="url">Allemagne</country>
</affiliation>
</author>
<author><name sortKey="Klee, Victor" sort="Klee, Victor" uniqKey="Klee V" first="Victor" last="Klee">Victor Klee</name>
<affiliation wicri:level="2"><country xml:lang="fr">États-Unis</country>
<placeName><region type="state">Washington (État)</region>
</placeName>
<wicri:cityArea>Department of Mathematics University of Washington Seattle</wicri:cityArea>
</affiliation>
<affiliation wicri:level="1"><country wicri:rule="url">Allemagne</country>
</affiliation>
</author>
<author><name sortKey="Tam, Bit Shun" sort="Tam, Bit Shun" uniqKey="Tam B" first="Bit-Shun" last="Tam">Bit-Shun Tam</name>
<affiliation wicri:level="1"><country xml:lang="fr">République populaire de Chine</country>
<wicri:regionArea>Department of Mathematics Tamkang University Tamsui, Taiwan 25137</wicri:regionArea>
<wicri:noRegion>Taiwan 25137</wicri:noRegion>
</affiliation>
<affiliation wicri:level="1"><country wicri:rule="url">Allemagne</country>
</affiliation>
</author>
</analytic>
<monogr></monogr>
<series><title level="j">Linear Algebra and Its Applications</title>
<title level="j" type="abbrev">LAA</title>
<idno type="ISSN">0024-3795</idno>
<imprint><publisher>ELSEVIER</publisher>
<date type="published" when="1995">1995</date>
<biblScope unit="volume">223–224</biblScope>
<biblScope unit="supplement">C</biblScope>
<biblScope unit="page" from="285">285</biblScope>
<biblScope unit="page" to="305">305</biblScope>
</imprint>
<idno type="ISSN">0024-3795</idno>
</series>
<idno type="istex">070F48080B858C2F6DCB61E9A4F48C7723DD1653</idno>
<idno type="DOI">10.1016/0024-3795(93)00364-6</idno>
<idno type="PII">0024-3795(93)00364-6</idno>
</biblStruct>
</sourceDesc>
<seriesStmt><idno type="ISSN">0024-3795</idno>
</seriesStmt>
</fileDesc>
<profileDesc><textClass></textClass>
<langUsage><language ident="en">en</language>
</langUsage>
</profileDesc>
</teiHeader>
<front><div type="abstract" xml:lang="en">For a closed, pointed n-dimensional convex cone K in Rn, let π(K) denote the set of all n × n real matrices A which as linear operators map K into itself. Let ∑(K) denote the set of all n × n matrices that are cross-positive on K, and L(K) = ∑(K) ∩ [− ∑(K)], the lineality space of ∑(K). Let Λ = RI, the set of all real multiples of the n × n identity matrix I. Then π(K)+Δ⊆π(K)+L(K)⊆cl[π(K)+Δ]=Σ(K). The final equality was proved in 1970 by Schneider and Vidyasagar, who showed also that π(K) + Λ = ∑(K) when K is polyhedral but not when K is a three-dimensional circular cone. They asked for a general characterization of those K for which the equality holds. It is shown here that if n ⩾ 3 and the cone K is strictly convex or smooth, then π(K) + Λ ≠ ∑(K); hence for n ⩾ 3 the equality fails for “almost all” K in the sense of Baire category. However, the equality does hold for some nonpolyhedral K, as was shown by a construction that appeared in the third author's 1977 dissertation and is explained here in more detail. In 1994 it was shown by Stern and Wolkowicz that the weaker equality π(K) + L(K) = ∑(K) holds for all ellipsoidal (as well as for all polyhedral) K, and they wondered whether this equality holds for all K. However, their equality certainly fails for all strictly convex or smooth K such that L(K) = Λ, and it is shown here that this also includes “almost all” K when n ⩾ 3.</div>
</front>
</TEI>
<affiliations><list><country><li>Allemagne</li>
<li>République populaire de Chine</li>
<li>États-Unis</li>
</country>
<region><li>Washington (État)</li>
</region>
</list>
<tree><country name="Allemagne"><noRegion><name sortKey="Gritzmann, Peter" sort="Gritzmann, Peter" uniqKey="Gritzmann P" first="Peter" last="Gritzmann">Peter Gritzmann</name>
</noRegion>
<name sortKey="Gritzmann, Peter" sort="Gritzmann, Peter" uniqKey="Gritzmann P" first="Peter" last="Gritzmann">Peter Gritzmann</name>
<name sortKey="Klee, Victor" sort="Klee, Victor" uniqKey="Klee V" first="Victor" last="Klee">Victor Klee</name>
<name sortKey="Tam, Bit Shun" sort="Tam, Bit Shun" uniqKey="Tam B" first="Bit-Shun" last="Tam">Bit-Shun Tam</name>
</country>
<country name="États-Unis"><region name="Washington (État)"><name sortKey="Klee, Victor" sort="Klee, Victor" uniqKey="Klee V" first="Victor" last="Klee">Victor Klee</name>
</region>
</country>
<country name="République populaire de Chine"><noRegion><name sortKey="Tam, Bit Shun" sort="Tam, Bit Shun" uniqKey="Tam B" first="Bit-Shun" last="Tam">Bit-Shun Tam</name>
</noRegion>
</country>
</tree>
</affiliations>
</record>
Pour manipuler ce document sous Unix (Dilib)
EXPLOR_STEP=$WICRI_ROOT/Wicri/Rhénanie/explor/UnivTrevesV1/Data/Main/Exploration
HfdSelect -h $EXPLOR_STEP/biblio.hfd -nk 002910 | SxmlIndent | more
Ou
HfdSelect -h $EXPLOR_AREA/Data/Main/Exploration/biblio.hfd -nk 002910 | SxmlIndent | more
Pour mettre un lien sur cette page dans le réseau Wicri
{{Explor lien |wiki= Wicri/Rhénanie |area= UnivTrevesV1 |flux= Main |étape= Exploration |type= RBID |clé= ISTEX:070F48080B858C2F6DCB61E9A4F48C7723DD1653 |texte= Cross-positive matrices revisited }}
![]() | This area was generated with Dilib version V0.6.31. | ![]() |