## 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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### References:

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