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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