  The $k$-matching polytope of a graph is the convex hull of all its matchings
of a given size $k$ when they are considered as indicator vectors. In this
paper, we prove that the $k$-matching polytope of a bipartite graph is normal,
that is, every integer point in its $t$-dilate is the sum of $t$ integers
points of the original polytope. This generalizes the known fact that Birkhoff
polytopes are normal. As a preliminary result, we prove that for bipartite
graphs the $k$-matching polytope is equal to the fractional $k$-matching
polytope, having thus the $H$-representation of the polytope. This generalizes
the Birkhoff-Von Neumann Theorem which establish that every doubly stochastic
matrix can be written as a convex combination of permutation matrices.
