It is proved that, for real \(n\)-vectors \(x\) and \(y\), \(x\) is majorized by \(y\) if and only if \(x = PHQy\) for some permutation matrices \(P,Q\) and for some doubly stochastic matrix \(H\) which is a direct sum of doubly stochastic Hessenberg matrices. This result reveals that any \(n\)-vector which is majorized by a vector \(y\) can be expressed as a convex combination of at most \((n^2 - n + 2)/2\) permutations of \(y\).