31. Permanent versus determinant over a finite fieldGregor Dolinar, Aleksandr Èmilevič Guterman, Bojan Kuzma, Marko Orel, 2013, published scientific conference contribution Abstract: 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. Keywords: mathematics, linear algebra, matrix theory, permanent, determinant Published in RUP: 03.04.2017; Views: 2030; Downloads: 123 Link to full text |
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36. Beasley, Leroy B.; Guterman, Alexander E.; Jun, Young-Bae; Song, Seok-Zun: Linear preservers of extremes of rank inequalities over semirings: Row and column ranks. (English). - [J] Linear Algebra Appl. 413, No. 2-3, 495-509 (2006). [ISSN 0024-3795]Bojan Kuzma, 2006, review, book review, critique Keywords: matematika, linearna algebra, lnearni ohranjevalci, polkolobar Published in RUP: 15.10.2013; Views: 2409; Downloads: 34 Link to full text |
37. Purificação Coelho, M.; Duffner, M.Antónia: Immanant preserving and immanant converting maps. (English). - [J] Linear Algebra Appl. 418, No. 1, 177-187 (2006). [ISSN 0024-3795]Bojan Kuzma, 2006, review, book review, critique Keywords: matematika, linearna algebra, linearne transformacije, ohranjevalci Published in RUP: 15.10.2013; Views: 2638; Downloads: 35 Link to full text |
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39. Li, Jinxiu: Applications of Wigner's theorem to positive maps preserving norm of operator products. (English). - [J] J. Math. Study 37, No. 1, 11-16 (2004). [ISSN 1006-6837]Bojan Kuzma, 2006, review, book review, critique Keywords: matematika, teorija operatorjev, Wignerjev izrek, operatorska algebra, norma, singularna vrednost Published in RUP: 15.10.2013; Views: 3389; Downloads: 40 Link to full text |
40. Noninvertibility preservers on Banach algebrasBojan Kuzma, 2006, short scientific article Abstract: It is proved that a linear surjection ▫$\Phi: \mathcal{A} \to \mathcal{B}$▫, which preserves noninvertibility between semisimple, unital, complex Banach algebras, is automatically injective. Keywords: mathematics, functional analysis, linear preserver, noninvertible element, semisimple Banach algebra, socle Published in RUP: 15.10.2013; Views: 3028; Downloads: 126 Link to full text |