| Title: | Permanent versus determinant over a finite field |
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| Authors: | ID Dolinar, Gregor (Author)
ID Guterman, Aleksandr Èmilevič (Author)
ID Kuzma, Bojan (Author)
ID Orel, Marko (Author) |
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| Files: |  http://dx.doi.org/10.1007/s10958-013-1469-4
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| Language: | English |
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| Work type: | Not categorized |
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| Typology: | 1.08 - Published Scientific Conference Contribution |
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| Organization: | IAM - Andrej Marušič Institute
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| 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. |
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| Keywords: | mathematics, linear algebra, matrix theory, permanent, determinant |
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| Year of publishing: | 2013 |
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| Number of pages: | Str. 404-413 |
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| PID: | 20.500.12556/RUP-7737  |
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| ISSN: | 1072-3374 |
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| UDC: | 512.643 |
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| COBISS.SI-ID: | 16715865 |
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| Publication date in RUP: | 03.04.2017 |
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| Views: | 2124 |
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| Downloads: | 125 |
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