Editing Permutation matrix (section)

==Doubly stochastic matrices==

Every permutation matrix is [[doubly stochastic matrix|doubly stochastic]].  The set of all doubly stochastic matrices is called the [[Birkhoff polytope]], and the permutation matrices play a special role in that polytope.  The [[Birkhoff–von Neumann theorem]] says that every doubly stochastic real matrix is a [[convex combination]] of permutation matrices of the same order, with the permutation matrices being precisely the [[extreme point]]s (the vertices) of the Birkhoff polytope. The Birkhoff polytope is thus the [[convex hull]] of the permutation matrices.<ref name=Bru19>{{harvnb|Brualdi|2006|p=19}}</ref>