Proving that a doubly stochastic matrix is a convex combination of permutation matrices

Asked Feb 28 '19 at 20:21

Active Feb 28 '19 at 20:21

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This is a homework problem in my graph theory class and I wanted tips on how to solve it. Any tips would be appreciated..

Show that any doubly stochastic matrix is a convex combination of permutation matrices, i.e. any double stochastic A can be written as A=t₁P₁ +t₂ +⋯t_lP_l, where each P_i is a permutation matrix, and the numbers t_i satisfy t₁, . . . , t_l ≥ 0 and ∑^l_i=1 t_i = 1.

asked Feb 28 '19 at 20:21

Davyd Rickmann

2

What you're trying to prove is known as the Birkhoff-von Neumann theorem. The standard proof of this uses Hall's matching theorem. Have you covered Hall's matching theorem in your graph theory class? – Ben Grossmann Feb 28 '19 at 20:31
Yes, thank you. I was able to figure it out! – Davyd Rickmann Mar 01 '19 at 03:34

Proving that a doubly stochastic matrix is a convex combination of permutation matrices

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