A set is convex if:Originally Posted by suedenation
Now if is the space of all doubly stochastic matrices,
and , then if:
then is doubly stochastic, since:
and:
Hence is convex.
RonL
An nxn matrix is called a permutation matrix if there is exactly one 1 in each row and column, and all other entries are 0.
An nxn matrix A=(aij) is called doubly stochastic if aij>=0 for all 1<=i,j<=n, and the sum of the entries in any row or column is 1.
Let C denote the set of all doubly stochastic nxn matrices.
(a) SHow that C is convex.
(b) Show that if P is an nxn permutation matrix, then P is an extreme point of the convex set C(the converse is also true).
Can anyone help? I have spent like 4 hours on this question and am going crazy. Thanks very much guys.
1. Clearly is a doubly stochastic matrix.Originally Posted by suedenation
2. For to be an extreme point of the convex set means that:
That is does not lie strictly between two points in .
(this is one definition; yours may be different but it should be equivalent)
From this you should be able to prove that is an extreme point of
RonL