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Determinant preserving transformations on symmetric matrix spaces. (English) Zbl 1069.15004
Let $$S_n(F)$$ be the vector space of $$n\times n$$ symmetric matrices over a field $$F$$ (with certain restrictions on cardinality and characteristic). The transformations $$\phi$$ on the space which satisfy one of the following conditions:
1. $$\text{det}(A +\lambda B)= \text{det}(\phi(A) + \lambda\phi(B))$$ for all $$A,B\in S_n(F)$$ and $$\lambda\in F$$;
2. $$\phi$$ is surjective and $$\text{det}(A+\lambda B)= \text{det}(\phi(A)+ \lambda\phi(B))$$ for all $$A$$, $$B$$ and two specific $$\lambda$$;
3. $$\phi$$ is additive and preserves determinant;
are characterized. The authors study determinant preservers on the vector space of symmetric matrices.

##### MSC:
 15A04 Linear transformations, semilinear transformations 15A03 Vector spaces, linear dependence, rank, lineability 15A15 Determinants, permanents, traces, other special matrix functions
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