Induced maps on matrices over fields. (English) Zbl 1474.15008

Summary: Suppose that \(\mathbb{F}\) is a field and \(m, n \geq 3\) are integers. Denote by \(M_{m n}(\mathbb{F})\) the set of all \(m \times n\) matrices over \(\mathbb{F}\) and by \(M_n(\mathbb{F})\) the set \(M_{n n}(\mathbb{F})\). Let \(f_{i j}\) (\(i \in [1, m], j \in [1, n]\)) be functions on \(\mathbb{F}\), where \([1, n]\) stands for the set \(\{1, \ldots, n \}\). We say that a map \(f : M_{m n}(\mathbb{F}) \rightarrow M_{m n}(\mathbb{F})\) is induced by \(\{f_{i j} \}\) if \(f\) is defined by \(f : [a_{i j}] \mapsto [f_{i j}(a_{i j})]\). We say that a map \(f\) on \(M_n(\mathbb{F})\) preserves similarity if \(A \sim B \Rightarrow f(A) \sim f(B)\), where \(A \sim B\) represents that \(A\) and \(B\) are similar. A map \(f\) on \(M_n(\mathbb{F})\) preserving inverses of matrices means \(f(A) f(A^{- 1}) = I_n\) for every invertible \(A \in M_n(\mathbb{F})\). In this paper, we characterize induced maps preserving similarity and inverses of matrices, respectively.


15A03 Vector spaces, linear dependence, rank, lineability
47B47 Commutators, derivations, elementary operators, etc.
Full Text: DOI


[1] Kalinowski, J., On rank equivalence and rank preserving operators, Novi Sad Journal of Mathematics, 32, 1, 133-139 (2002) · Zbl 1274.15001
[2] Kalinowski, J., On functions preserving rank of matrices, Mathematical Notes. A Publication of the University of Miskolc, 4, 1, 35-37 (2003) · Zbl 1026.15001
[3] Liu, S.-W.; Zhang, G.-D., Maps preserving rank 1 matrices over fields, Journal of Natural Science of Heilongjiang University, 23, 1, 138-140 (2006) · Zbl 1110.15006
[4] Du, S.; Hou, J.; Bai, Z., Nonlinear maps preserving similarity on \(B(H)\), Linear Algebra and its Applications, 422, 2-3, 506-516 (2007) · Zbl 1124.47025 · doi:10.1016/j.laa.2006.11.008
[5] Oikhberg, T.; Peralta, A. M., Automatic continuity of orthogonality preservers on a non-commutative \(L^p(\tau)\) space, Journal of Functional Analysis, 264, 8, 1848-1872 (2013) · Zbl 1288.47034 · doi:10.1016/j.jfa.2013.01.019
[6] Chebotar, M. A.; Ke, W.-F.; Lee, P.-H.; Wong, N.-C., Mappings preserving zero products, Studia Mathematica, 155, 1, 77-94 (2003) · Zbl 1032.46063 · doi:10.4064/sm155-1-6
[7] Leung, C.-W.; Ng, C.-K.; Wong, N.-C., Linear orthogonality preservers of Hilbert C*-modules over C*-algebras with real rank zero, Proceedings of the American Mathematical Society, 140, 9, 3151-3160 (2012) · Zbl 1282.46050 · doi:10.1090/S0002-9939-2012-11260-2
[8] Konnov, I. V.; Yao, J. C., On the generalized vector variational inequality problem, Journal of Mathematical Analysis and Applications, 206, 1, 42-58 (1997) · Zbl 0878.49006 · doi:10.1006/jmaa.1997.5192
[9] Huo, D.; Zheng, B.; Liu, H., Characterizations of nonlinear Lie derivations of \(B(X)\), Abstract and Applied Analysis, 2013 (2013) · Zbl 1271.47029 · doi:10.1155/2013/245452
[10] Yao, H.; Zheng, B., Zero triple product determined matrix algebras, Journal of Applied Mathematics, 2012 (2012) · Zbl 1239.16031 · doi:10.1155/2012/925092
[11] Xu, J.; Zheng, B.; Yao, H., Linear transformations between multipartite quantum systems that map the set of tensor product of idempotent matrices into idempotent matrix set, Journal of Function Spaces and Applications, 2013 (2013) · Zbl 1446.15002 · doi:10.1155/2013/182569
[12] Xu, J.; Zheng, B.; Yang, L., The automorphism group of the Lie ring of real skew-symmetric matrices, Abstract and Applied Analysis, 2013 (2013) · Zbl 1470.17011 · doi:10.1155/2013/638230
[13] Song, X.; Cao, C.; Zheng, B., A note on \(k\)-potence preservers on matrix spaces over complex field, Abstract and Applied Analysis, 2013 (2013) · Zbl 1291.15069 · doi:10.1155/2013/581683
[14] Zheng, B.; Xu, J.; Fošner, A., Linear maps preserving rank of tensor products of matrices, Linear and Multilinear Algebra (2014) · Zbl 1310.15048 · doi:10.1080/03081087.2013.869589
[15] Yang, L.; Zhang, W.; Xu, J. L., Linear maps on upper triangular matrices spaces preserving idempotent tensor products, Abstract and Applied Analysis, 2014 (2014) · Zbl 1470.15024 · doi:10.1155/2014/148321
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