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help me

  • Subject: [mg3393] help me
  • From: ruivo at (Joaquim Azevedo)
  • Date: 3 Mar 1996 09:19:15 -0600
  • Approved:
  • Distribution: local
  • Newsgroups: wri.mathgroup
  • Organization: Wolfram Research, Inc.
  • Sender: daemon at

I am a mathematical student. In this moment I study graph  theory, but I 
have some problems to answer the question above. 
I would like some help, because I can not get contact with my teacher, 
and I have examination in two days. Please answer me.     

	A non-negative real matrix Q is doubly stochastic if the sum of the
entries in each row of Q is 1 and sum of the entries in each column of Q  is 1.
 A Permutation matrix is a (0,1)-matrix which has exactly one 1 in each row 
and each column.
 (Thus every permutation matrix is doubly stochastic.)  Show that:

(a)  every doubly stochastic matrix is necessarily square;

(b)  every doubly stochastic matrix Q can be expressed as convex linear 
     combination of permutation matrices; that is

 Q = c1P1 + c2P2 + ... + ckPk
   where each Pi  is a permutation matrix, each ci is a non-negative real 
   number, and ( ci = 1 (i=1,...,k)

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