{"?xml":{"@version":"1.0"},"edm:RDF":{"@xmlns:dc":"http://purl.org/dc/elements/1.1/","@xmlns:edm":"http://www.europeana.eu/schemas/edm/","@xmlns:wgs84_pos":"http://www.w3.org/2003/01/geo/wgs84_pos","@xmlns:foaf":"http://xmlns.com/foaf/0.1/","@xmlns:rdaGr2":"http://rdvocab.info/ElementsGr2","@xmlns:oai":"http://www.openarchives.org/OAI/2.0/","@xmlns:owl":"http://www.w3.org/2002/07/owl#","@xmlns:rdf":"http://www.w3.org/1999/02/22-rdf-syntax-ns#","@xmlns:ore":"http://www.openarchives.org/ore/terms/","@xmlns:skos":"http://www.w3.org/2004/02/skos/core#","@xmlns:dcterms":"http://purl.org/dc/terms/","edm:WebResource":[{"@rdf:about":"http://www.dlib.si/stream/URN:NBN:SI:doc-E104WZ4F/95713162-12b1-4f79-9d79-ec8b83958f50/PDF","dcterms:extent":"447 KB"},{"@rdf:about":"http://www.dlib.si/stream/URN:NBN:SI:doc-E104WZ4F/6de4d218-9ca5-4834-9f33-84d7b2dd6df1/TEXT","dcterms:extent":"34 KB"}],"edm:TimeSpan":{"@rdf:about":"2008-2025","edm:begin":{"@xml:lang":"en","#text":"2008"},"edm:end":{"@xml:lang":"en","#text":"2025"}},"edm:ProvidedCHO":{"@rdf:about":"URN:NBN:SI:doc-E104WZ4F","dcterms:isPartOf":[{"@rdf:resource":"https://www.dlib.si/details/URN:NBN:SI:spr-UP1WMFAR"},{"@xml:lang":"sl","#text":"Ars mathematica contemporanea"}],"dcterms:issued":"2021","dc:creator":"Grinberg, Darij","dc:format":[{"@xml:lang":"sl","#text":"številka:2"},{"@xml:lang":"sl","#text":"letnik:20"},{"@xml:lang":"sl","#text":"str. 261-274"}],"dc:identifier":["DOI:10.26493/1855-3974.2248.d3f","ISSN:1855-3966","COBISSID_HOST:92615427","URN:URN:NBN:SI:doc-E104WZ4F"],"dc:language":"en","dc:publisher":{"@xml:lang":"sl","#text":"Univerza na Primorskem, Fakulteta za matematiko, naravoslovje in informacijske tehnologije"},"dc:subject":[{"@xml:lang":"en","#text":"compound matrix"},{"@xml:lang":"en","#text":"determinant"},{"@xml:lang":"sl","#text":"determinanta"},{"@xml:lang":"sl","#text":"polinomi"},{"@xml:lang":"en","#text":"polynomials"},{"@xml:lang":"sl","#text":"sestavljena matrika"},{"@xml:lang":"en","#text":"Sylvester's determinant"},{"@xml:lang":"sl","#text":"Sylvestrova determinanta"}],"dcterms:temporal":{"@rdf:resource":"2008-2025"},"dc:title":{"@xml:lang":"sl","#text":"A double Sylvester determinant|"},"dc:description":[{"@xml:lang":"sl","#text":"Given two ?$\\left( n+1\\right) \\times\\left( n+1\\right)$?-matrices ?$A$? and ?$B$? over a commutative ring, and some ?$k\\in\\left\\{ 0,1,\\ldots,n\\right\\}$?, we consider the ?$\\dbinom{n}{k}\\times\\dbinom{n}{k}$?-matrix ?$W$? whose entries are ?$\\left( k+1\\right) \\times\\left( k+1\\right)$?-minors of ?$A$? multiplied by corresponding ?$\\left( k+1\\right) \\times\\left( k+1\\right)$?-minors of ?$B$?. Here we require the minors to use the last row and the last column (which is why we obtain an ?$\\dbinom{n}{k}\\times\\dbinom{n}{k}$?-matrix, not an ?$\\dbinom{n+1}{k+1}\\times\\dbinom{n+1}{k+1}$?-matrix). We prove that the determinant ?$\\det W$? is a multiple of ?$\\det A$? if the ?$\\left( n+1,n+1\\right)$?-th entry of ?$B$? is ?$0$?. Furthermore, if the ?$\\left( n+1,n+1\\right)$?-th entries of both ?$A$? and ?$B$? are ?$0$?, then ?$\\det W$? is a multiple of ?$\\left( \\det A\\right) \\left( \\det B\\right)$?. This extends a previous result of Olver and the author"},{"@xml:lang":"sl","#text":"Če sta dani dve ?$\\left( n+1\\right) \\times\\left( n+1\\right)$?-matriki ?$A$? in ?$B$? nad komutativnim kolobarjem in neko število ?$k\\in\\left\\{ 0,1,\\ldots,n\\right\\}$?, obravnavamo ?$\\dbinom{n}{k}\\times\\dbinom{n}{k}$?-matriko ?$W$?, katere elementi so ?$\\left( k+1\\right) \\times\\left( k+1\\right)$?-minorji matrike ?$A$?, pomnoženi z ustreznimi $?\\left( k+1\\right) \\times\\left( k+1\\right)$?-minorji matrike ?$B$?. Pri tem zahtevamo, da minorji vsebujejo zadnjo vrstico in zadnji stolpec (zato dobimo ?$\\dbinom{n}{k}\\times\\dbinom{n}{k}$?-matriko, ne pa ?$\\dbinom{n+1}{k+1} \\times \\dbinom{n+1}{k+1}$?-matrike). Dokažemo, da je determinanta ?$\\det W$? večkratnik determinante ?$\\det A$?, če je? $\\left( n+1,n+1\\right)$?-ti element matrike ?$B$? enak ?$0$?. Nadalje dokažemo, da če sta ?$\\left( n+1,n+1\\right)$?-ta elementa tako matrike ?$A$? kot matrike ?$B$? enaka ?$0$?, potem je ?$\\det W$? večkratnik od ?$\\left( \\det A\\right) \\left( \\det B\\right)$?. To predstavlja razširitev prejšnjega rezultata Olverja in avtorja tega članka"}],"edm:type":"TEXT","dc:type":[{"@xml:lang":"sl","#text":"znanstveno časopisje"},{"@xml:lang":"en","#text":"journals"},{"@rdf:resource":"http://www.wikidata.org/entity/Q361785"}]},"ore:Aggregation":{"@rdf:about":"http://www.dlib.si/?URN=URN:NBN:SI:doc-E104WZ4F","edm:aggregatedCHO":{"@rdf:resource":"URN:NBN:SI:doc-E104WZ4F"},"edm:isShownBy":{"@rdf:resource":"http://www.dlib.si/stream/URN:NBN:SI:doc-E104WZ4F/95713162-12b1-4f79-9d79-ec8b83958f50/PDF"},"edm:rights":{"@rdf:resource":"http://creativecommons.org/licenses/by/4.0/"},"edm:provider":"Slovenian National E-content Aggregator","edm:intermediateProvider":{"@xml:lang":"en","#text":"National and University Library of Slovenia"},"edm:dataProvider":{"@xml:lang":"sl","#text":"Univerza na Primorskem, Fakulteta za naravoslovje, matematiko in informacijske tehnologije"},"edm:object":{"@rdf:resource":"http://www.dlib.si/streamdb/URN:NBN:SI:doc-E104WZ4F/maxi/edm"},"edm:isShownAt":{"@rdf:resource":"http://www.dlib.si/details/URN:NBN:SI:doc-E104WZ4F"}}}}