Determinant preserving transformations on symmetric matrix spaces. (English) Zbl 1069.15004

Cao, Chongguang; Tang, Xiaomin

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)

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