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2. Commuting graphs and extremal centralizersGregor Dolinar, Aleksandr Èmilevič Guterman, Bojan Kuzma, Polona Oblak, 2014, izvirni znanstveni članek Opis: We determine the conditions for matrix centralizers which can guarantee the connectedness of the commuting graph for the full matrix algebra ▫$M_n(\mathbb{F})$▫ over an arbitrary field ▫$\mathbb{F}$▫. It is known that if ▫$\mathbb{F}$▫ is an algebraically closed field and ▫$n \ge 3$▫, then the diameter of the commuting graph of ▫$M_n(\mathbb{F})$▫ is always equal to four. We construct a concrete example showing that if ▫$\mathbb{F}$▫ is not algebraically closed, then the commuting graph of ▫$M_n(\mathbb{F})$▫ can be connected with the diameter at least five. Ključne besede: commuting graph, matrix ring, centralizer Objavljeno v RUP: 31.12.2021; Ogledov: 725; Prenosov: 22 Celotno besedilo (228,78 KB) |
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4. Bar´ery Gibsona dlja problemy PoliaGregor Dolinar, Aleksandr Èmilevič Guterman, Bojan Kuzma, 2010, objavljeni znanstveni prispevek na konferenci Opis: V članku je obravnavana spodnja meja za število neničelnih elementov v ▫$(0, 1)$▫ matrikah, pri katerem se da permanento vedno pretvoriti v determinanto samo s spreminjanjem predznaka ▫$pm$▫ elementom matrike. Ključne besede: matematika, linearna algebra, teorija matrik, permanenta, determinanta Objavljeno v RUP: 03.04.2017; Ogledov: 2007; Prenosov: 64 Povezava na celotno besedilo |
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7. On maximal distances in a commuting graphGregor Dolinar, Bojan Kuzma, Polona Oblak, 2012, izvirni znanstveni članek Opis: It is shown that matrices over algebraically closed fields that are farthest apart in the commuting graph must be non-derogatory. Rank-one matrices and diagonalizable matrices are also characterized in terms of the commuting graph. Ključne besede: matematika, linearna algebra, teorija grafov, komutirajoči grafi, matrična algebra, algebraično zaprt obseg, centralizator, razdalja v grafih, mathematics, linear algebra, graph theory, commuting graph, matrix algebra, algebraically closed field, centralizer, distance in graphs Objavljeno v RUP: 03.04.2017; Ogledov: 2227; Prenosov: 256 Povezava na celotno besedilo |
8. Permanent versus determinant over a finite fieldGregor Dolinar, Aleksandr Èmilevič Guterman, Bojan Kuzma, Marko Orel, 2013, objavljeni znanstveni prispevek na konferenci Opis: Let ▫$\mathbb{F}$▫ be a finite field of characteristic different from 2. We study the cardinality of sets of matrices with a given determinant or a given permanent for the set of Hermitian matrices ▫$\mathcal{H}_n(\mathbb{F})$▫ and for the whole matrix space ▫$M_n(\mathbb{F})$▫. It is known that for ▫$n = 2$▫, there are bijective linear maps ▫$\Phi$▫ on ▫$\mathcal{H}_n(\mathbb{F})$▫ and ▫$M_n(\mathbb{F})$▫ satisfying the condition per ▫$A = \det \Phi(A)$▫. As an application of the obtained results, we show that if ▫$n \ge 3$▫, then the situation is completely different and already for ▫$n = 3$▫, there is no pair ofmaps ▫$(\Phi, \phi)$▫, where ▫$\Phi$▫ is an arbitrary bijective map on matrices and ▫$\phi \colon \mathbb{F} \to \mathbb{F}$▫ is an arbitrary map such that per ▫$A = \phi(\det \Phi(A))$▫ for all matrices ▫$A$▫ from the spaces ▫$\mathcal{H}_n(\mathbb{F})$▫ and ▫$M_n(\mathbb{F})$▫, respectively. Moreover, for the space ▫$M_n(\mathbb{F})$▫, we show that such a pair of transformations does not exist also for an arbitrary ▫$n > 3$▫ if the field ▫$\mathbb{F}$▫ contains sufficiently many elements (depending on ▫$n$▫). Our results are illustrated by a number of examples. Ključne besede: mathematics, linear algebra, matrix theory, permanent, determinant Objavljeno v RUP: 03.04.2017; Ogledov: 2028; Prenosov: 123 Povezava na celotno besedilo |
9. Maps on self-adjoint operators preserving numerical range of products up to a factorKan He, Jin Chuan Hou, Gregor Dolinar, Bojan Kuzma, 2011, izvirni znanstveni članek Opis: Let ▫$H$▫ be a complex Hilbert space and ▫${mathscr{S}}_a(H)$▫ the space of all self adjoint operators on ▫$H$▫. ▫$Phi colon {mathscr{S}}_a(H) to {mathscr{S}}_a(H)$▫ is a surjective map. For ▫$xi, eta in mathbb{C} setminus {1}$▫, then ▫$Phi$▫ satisfies that ▫$$W(AB - xi BA) = W(Phi(A)Phi(B) - etaPhi(B)phi(A))$$▫ for all ▫$A,B in {mathscr{S}}_a(H)$▫ if and only if there exists a unitary operator or con-unitary operator ▫$U$▫ such that ▫$Phi(A) = UAU^ast$▫ for all ▫$A in {mathscr{S}}_a(H)$▫ or ▫$Phi(A) = -UAU^ast$▫ for all ▫$A in {mathscr{S}}_a(H)$▫. Ključne besede: matematika, teorija operatorjev, numerični zaklad, ohranjevalci Objavljeno v RUP: 03.04.2017; Ogledov: 2251; Prenosov: 32 Povezava na celotno besedilo |