Linear operators preserving the minimal rank
Properties of the minimal rank \(r(X)\) defined as min\(\{r(X-\lambda I):\lambda \in \mathbb{F}\}\), where \(X\) belongs to the set \(\mathbb{F}^{n \times n}\) of \(n \times n\) matrices over an algebraically closed field \(\mathbb{F}\) and \(I\) is the \(n \times n\) identity matrix, are studied. The forms of linear operators for \(n \geq 3\) and \(n = 2\) preserving the minimal rank or equivalently the number of nontrivial invariant polynomials are described. The existence of linear operators converting the minimal rank to the ordinary rank is also investigated.
scientific article; zbMATH DE number 1419881