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2.
Commuting graphs and extremal centralizers
Gregor 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: 930; Prenosov: 23
.pdf Celotno besedilo (228,78 KB)

3.
Bar´ery Gibsona dlja problemy Polia
Gregor 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: 2211; Prenosov: 64
URL Povezava na celotno besedilo

4.
Permanent versus determinant over a finite field
Gregor 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: 2249; Prenosov: 127
URL Povezava na celotno besedilo

5.
Rank-permutable additive mappings
Anna A. Alieva, Aleksandr Èmilevič Guterman, Bojan Kuzma, 2006, izvirni znanstveni članek

Opis: Let ▫$\sigma$▫ be a fixed non-identical permutation on ▫$k$▫ elements. Additive bijections ▫$T$▫ on the matrix algebra ▫$M_n(\mathbb{F})$▫ over a field ▫$\mathbb{F}$▫ of characteristic zero, with the property that ▫$\rm{rk} (A_1...A_k) = \rm{rk} (A_{\sigma(1)}...A_{\sigma(k)})$▫ implies the same condition on the ▫$T$▫ images, are characterized. It is also shown that the surjectivity assumption can be relaxed, if this property is preserved in both directions.
Ključne besede: mathematics, linearna algebra, matrix algebra, rank, permutation, additive preservers
Objavljeno v RUP: 15.10.2013; Ogledov: 3715; Prenosov: 89
URL Povezava na celotno besedilo

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