Cao, Chongguang; Tang, Xiaomin Determinant preserving transformations on symmetric matrix spaces. (English) Zbl 1069.15004 Electron. J. Linear Algebra 11, 205-211 (2004). 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. Reviewer: Yueh-er Kuo (Knoxville) Cited in 8 Documents MSC: 15A04 Linear transformations, semilinear transformations 15A03 Vector spaces, linear dependence, rank, lineability 15A15 Determinants, permanents, traces, other special matrix functions Keywords:linear preserving problem; rank; symmetric matrix; determinant preserving transformations PDF BibTeX XML Cite \textit{C. Cao} and \textit{X. Tang}, Electron. J. Linear Algebra 11, 205--211 (2004; Zbl 1069.15004) Full Text: DOI EMIS EuDML