A generalization of Sourour's theorem
\textit{Sourour} [Linear Multilinear Algebra 19, 141-147 (1986; Zbl 0591.15008)] proved the following theorem: Let \(A\) be a nonscalar invertible matrix over a field \(K\), then there exist matrices \(B\), \(C\) with prescribed eigenvalues such that \(A=BC\). This result can be reformulated in the following two ways: (A) For every noncentral invertible matrix there is a matrix in its \(SL_n\)-conjugacy class which has a Gauss decomposition with prescribed diagonal part, and (B) For every noncentral matrix \(X\in GL_n (K)\), \(n>1\), and every sequence \((a_1, \dots, a_{n-1})\), \(a_i\in K\setminus 0\), there is a matrix \(Y\in SL_n(K)\) such that the \(i\)-th principal minor of \(YXY^{-1}\) has the value \(a_i\), \(1\leq i<n\). (A) was extended to Chevalley groups by the authors, and the purpose of this note is to remove the restriction ``\(a_i\neq 0\) for all \(i\)'' in (B).
scientific article; zbMATH DE number 1331732