How to find by hand or by software a doubly stochastic matrix? See the picture uploaded with this message. Thank a lot.
Theorem(Birkhoff). Every doubly stochastic matrix is a convex combination of permutation matrices.
I suggest you to take a look at the following book:
R.A. Horn, and C.R. Johnson, Matrix Analysis, Cambridge University Press, Cambridge, 1996. (Theorem 8.7.1 pp.527-528)
Hello,
take a look to the following link it maybe helpful:
https://www.mathworks.com/matlabcentral/answers/53957-is-there-a-better-way-to-randomly-generate-a-doubly-stochastic-matrix?requestedDomain=www.mathworks.com
@ Zoran Jasak I don't have this software. Can you use this software, please find that double random matrix P, thank you!
Dear Feng, why don't try with your hypothesis above for a a doubly stochastic matrix with small orders 2, 3, may be you can find a pattern.
For n=2, I just was calculating, P=[p1 1-p1 , 1-p1 p1] with p1>0
and (1-p1) / p1 =-x2 / x1
This problem proposed by Professor Qi Feng from the book《MARSHALL A W, OLKIN I, ARNOLD B C. Inequalities: theory of majorization and its application (Second Edition) [M]. New York : Springer Press, 2011.》pp.295-296.
The ssociate professor Wang Dongsheng gets this double random matrix find by hand.
The question should be corrected as the one showed in the picture uploaded in this message. The answer should be the one at https://www.researchgate.net/file.PostFileLoader.html?id=5786c728eeae39193e6c6525&assetKey=AS%3A383523769864192%401468450600742
Is it right, Professor Shi?
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