Linear maps preserving permutation and stochastic matrices (Q5957179)

A linear transformation \(T\) of \(\mathbb{R}^{m\times n}\) into itself is a ``linear preserver'' of a subset \(\mathcal{S}\subseteq\mathbb{R}^{m\times n}\) if \(T(\mathcal{S)}\subseteq\mathcal{S}\), and \(T\) is a ``strong linear preserver'' if \(T(\mathcal{S)}=\mathcal{S}.\) Let \(DS(n)\) be the set of doubly stochastic matrices in \(\mathbb{R}^{n\times n}\) and \(CS(m,n)\) be the column stochastic matrices in \(\mathbb{R}^{m\times n}.\) The authors show that every strong linear preserver of \(DS(n)\) has the form \(X\mapsto PXQ\) or \(X\mapsto PX^{t}Q\) for some permutation matrices \(P\) and \(Q\), and every strong linear preserver of \(CS(m,n)\) has the form \(X\mapsto[P_{1}X_{1},\dots ,P_{n}X_{n}]Q\) where \(X_{1},\dots ,X_{n}\) are the columns of \(X\) and \(P_{1},\dots ,P_{n},Q\) are permutation matrices. Similar results are obtained for related sets such as the set of permutation matrices and the set of nonnegative matrices whose column sums are at most \(1\). A characterization of the linear preservers of \(DS(n)\) or \(CS(m,n)\) which are not necessarily strong linear preservers is much more complicated, but the authors are able to give a general description of the extremal elements (there are six distinct types).