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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.

MSC:

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