Applications of \(M\)-matrices to majorization (Q1187379)

It is known that for any two vectors \(x,y\) in the \(n\)-dimensional Euclidean space such that \(x_ 1\geq x_ 2\geq\cdots\geq x_ n\) and \(y_ 1\geq y_ 2\geq\cdots\geq y_ n\) the vector \(x\) is majorized by \(y\) if and only if \(x=Ay\) for some \(n\times n\) nonnegative definite doubly stochastic matrix \(A\). This paper uses \(M\)-matrices to characterize majorization and calculates an explicit formula for finding the matrix \(A\) defined above.