Quadratic transformations on matrices: Rank preservers
An algebraically closed field \(F\) of characteristic not 2 and two \(n \times n\) matrices, one being \(X = (X_{ij})\) with independent indeterminates over \(F\) as entries, the second \(Q(X) = (q_{ij} (X))\) with quadratic \(F\)-polynomials in the \(X_{ij}\) as entries, are considered. It is shown that for \(n \geq 5\), \(Q(X)\) satisfies: \(\text{rank} (A^2) = r\) implies \(\text{rank} (Q(A)) = r\) for all \(A \in F^{n \times n}\) for \(r = 0\), 1, and 2, if and only if there exist invertible matrices \(P_1\), \(P_2\) in \(F^{n \times n}\) such that either \(Q(X) = P_1 X^2P_2\) or \(Q(X) = P_1 (X^2)^t P_2\), where \((X^2)^t\) denote the transpose of \((X^2)\).
scientific article; zbMATH DE number 843879