Induced Maps on Matrices over Fields

Suppose that 𝔽 is a field and m,n≥3 are integers. Denote by Mmn(𝔽) the set of all m×n matrices over 𝔽 and by Mn(𝔽) the set Mnn(𝔽). Let fij (i∈[1,m],j∈[1,n]) be functions on 𝔽, where [1,n] stands for the set {1,…,n}. We say that a map f:Mmn(𝔽)→Mmn(𝔽) is induced by {fij} if f is defined by f:[aij]↦[fi...

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Main Authors: Li Yang, Xuezhi Ben, Ming Zhang, Chongguang Cao
Format: Article
Language:English
Published: Wiley 2014-01-01
Series:Abstract and Applied Analysis
Online Access:http://dx.doi.org/10.1155/2014/596756
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author Li Yang
Xuezhi Ben
Ming Zhang
Chongguang Cao
author_facet Li Yang
Xuezhi Ben
Ming Zhang
Chongguang Cao
author_sort Li Yang
collection DOAJ
description Suppose that 𝔽 is a field and m,n≥3 are integers. Denote by Mmn(𝔽) the set of all m×n matrices over 𝔽 and by Mn(𝔽) the set Mnn(𝔽). Let fij (i∈[1,m],j∈[1,n]) be functions on 𝔽, where [1,n] stands for the set {1,…,n}. We say that a map f:Mmn(𝔽)→Mmn(𝔽) is induced by {fij} if f is defined by f:[aij]↦[fij(aij)]. We say that a map f on Mn(𝔽) preserves similarity if A~B⇒f(A)~f(B), where A~B represents that A and B are similar. A map f on Mn(𝔽) preserving inverses of matrices means f(A)f(A-1)=In for every invertible A∈Mn(𝔽). In this paper, we characterize induced maps preserving similarity and inverses of matrices, respectively.
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spelling doaj-art-9a462aa1b3bb4a4586b114c6aa8e849c2025-02-03T05:46:45ZengWileyAbstract and Applied Analysis1085-33751687-04092014-01-01201410.1155/2014/596756596756Induced Maps on Matrices over FieldsLi Yang0Xuezhi Ben1Ming Zhang2Chongguang Cao3Department of Foundation, Harbin Finance University, Harbin 150030, ChinaSchool of Mathematical Science, Heilongjiang University, Harbin 150080, ChinaSchool of Mathematical Science, Heilongjiang University, Harbin 150080, ChinaSchool of Mathematical Science, Heilongjiang University, Harbin 150080, ChinaSuppose that 𝔽 is a field and m,n≥3 are integers. Denote by Mmn(𝔽) the set of all m×n matrices over 𝔽 and by Mn(𝔽) the set Mnn(𝔽). Let fij (i∈[1,m],j∈[1,n]) be functions on 𝔽, where [1,n] stands for the set {1,…,n}. We say that a map f:Mmn(𝔽)→Mmn(𝔽) is induced by {fij} if f is defined by f:[aij]↦[fij(aij)]. We say that a map f on Mn(𝔽) preserves similarity if A~B⇒f(A)~f(B), where A~B represents that A and B are similar. A map f on Mn(𝔽) preserving inverses of matrices means f(A)f(A-1)=In for every invertible A∈Mn(𝔽). In this paper, we characterize induced maps preserving similarity and inverses of matrices, respectively.http://dx.doi.org/10.1155/2014/596756
spellingShingle Li Yang
Xuezhi Ben
Ming Zhang
Chongguang Cao
Induced Maps on Matrices over Fields
Abstract and Applied Analysis
title Induced Maps on Matrices over Fields
title_full Induced Maps on Matrices over Fields
title_fullStr Induced Maps on Matrices over Fields
title_full_unstemmed Induced Maps on Matrices over Fields
title_short Induced Maps on Matrices over Fields
title_sort induced maps on matrices over fields
url http://dx.doi.org/10.1155/2014/596756
work_keys_str_mv AT liyang inducedmapsonmatricesoverfields
AT xuezhiben inducedmapsonmatricesoverfields
AT mingzhang inducedmapsonmatricesoverfields
AT chongguangcao inducedmapsonmatricesoverfields