An extension of Birkhoff's theorem with an application to determinants (Q2383022)

Let \(J\) be the \(n\times n\) diagonal matrix of the form \(\text{diag}(\undersetbrace n_+\to {+1,\dots,+ 1}, \undersetbrace n_-\to {-1,\dots,-1})\) with \(n= n_++ n_-\). With this matrix the authors associate a certain class of \(n\times n\) matrices, called \(J\)-doubly stochastic matrices, which in the special case \(n_+= n\) reduces to the class of doubly stochastic matrices, i.e., matrices with entries from \([0,1]\) and with row and column sums equal to one. They extend the Birkhoff theorem to this more general class of matrices.