Full column rank preservers that preserve semipositivity of matrices
For a field \(\mathbb{F}\), let \(M_{m ,n}(\mathbb{F})\) denote the set of all \(m \times n\) matrices over \(\mathbb{F}\). When \(m =n\), this set will be denoted by \(M_{n}(\mathbb{F})\). A matrix \(A \in M_{m ,n}(\mathbb{R})\) is said to be semipositive if there exists \(x \in \mathbb{R}^{n}\) with \(x \geq 0\) such that \(Ax >0\) where the inequalities are understood component-wise. If \(A \in M_{m ,n}(\mathbb{R})\) is semipositive and no proper \(m \times p\) submatrix of \(A\) is semipositive, then \(A\) is said to be minimally semipositive. A subspace \(\mathcal{V}\) of \(M_{n}(\mathbb{F})\) is called nonsingular if every nonzero element of \(\mathcal{V}\) is invertible. A linear map \(\phi :M_{n}(\mathbb{F}) \rightarrow M_{n}(\mathbb{F})\) preserves invertibility if \(\phi (A)\) is invertible whenever \(A\in M_{n}(\mathbb{F})\) is invertible. The following result was proved in [\textit{C. de Seguins Pazzis}, Linear Algebra Appl. 433, No. 2, 483--490 (2010; Zbl 1196.15029)]: Theorem. Let \(n \geq 2\) and let \(\phi :M_{n}(\mathbb{F}) \rightarrow M_{n}(\mathbb{F})\) be a linear unital invertibility preserving map. Then \(\phi \) is of the standard form, i.e., \(\phi (A) =TAT^{ -1}\) or \(\phi (A) =TA^{t}T^{ -1}\), \(A \in M_{n}(\mathbb{F})\), for some invertible matrix \(T \in M_{n}(\mathbb{F})\), or \(\phi \) is degenerate, i.e., there exists a \(n\)-dimensional nonsingular subspace \(\mathcal{V}\) of \(M_{n}(\mathbb{F})\), a bijective linear map \(\Phi :\mathbb{F}^{n} \rightarrow \mathcal{V}\) and a nonzero \(x \in \mathbb{F}^{n}\) such that \(\phi (A) =\Phi (Ax)\) for every \(A \in M_{n}(\mathbb{F})\) or \(\phi (A) =\Phi (A^{t}x)\) for every \(A \in M_{n}(\mathbb{F})\). Let us remark that \textit{L. Rodman} and \textit{P. Šemrl} independently studied a similar linear preserver problem and presented a generalization of the above result to \(M_{m ,n}(\mathbb{F})\) in [Linear Algebra Appl. 433, No. 11--12, 2257--2268 (2010; Zbl 1215.15031)] (using a different approach). In the paper under review, the authors first study preservers on \(M_{n}(\mathbb{R})\) of minimal semipositivity, i.e., transformations that map the subset of all minimally semipositive matrices in \(M_{n}(\mathbb{R})\) into itself. Using the above result, they prove that the only linear unital invertibility preserving map \(\phi :M_{n}(\mathbb{R}) \rightarrow M_{n}(\mathbb{R})\) that preserves minimal semipositivity is of the standard form, i.e., it can not be degenerate. Moreover, they show that if a linear map \(\phi :M_{2}(\mathbb{R}) \rightarrow M_{2}(\mathbb{R})\) given by \(A \mapsto TAT^{ -1}\) preserves minimal semipositivity, then \(T\) is necessarily monomial, and study the case when \(n >2\). The authors also prove that any linear unital invertibility preserving map \(\phi \) on \(M_{n}(\mathbb{R})\) that preserves semipositivity of matrices also can not be degenerate. Next, they study preservers of semipositivity and minimal semipositivity in a more general, i.e., rectangular case. As a corollary to the result of Rodman and Šemrl [loc. cit.], they show that if a linear map \(\phi :M_{m ,n}(\mathbb{R}) \rightarrow M_{m ,n}(\mathbb{R})\) is a full column rank preserver with \(m >n\) that preserves (minimal) semipositivity, then such a map can not be degenerate. The authors conclude by highlighting the structure of nonsingular subspaces of dimension \(2\) in \(M_{2}(\mathbb{R})\).
scientific article; zbMATH DE number 6890432