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: | , , , |
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| Format: | Article |
| Language: | English |
| Published: |
Wiley
2014-01-01
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| Series: | Abstract and Applied Analysis |
| Online Access: | http://dx.doi.org/10.1155/2014/596756 |
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| Summary: | 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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| ISSN: | 1085-3375 1687-0409 |