       help me

• To: mathgroup at smc.vnet.net
• Subject: [mg3393] help me
• From: Joaquim Azevedo <ruivo at grupo.bfe.pt>
• Date: Sun, 3 Mar 1996 02:25:06 -0500
• Sender: owner-wri-mathgroup at wolfram.com

```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.

Problem:
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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```

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