Cute random matrix result: Let p_n be the group of (n x n) - TopicsExpress

Cute random matrix result: Let p_n be the group of (n x n) matrices with a single 1 entry on each row and column, and zeros elsewhere. The matrices p_n are called the permutation matrices. They faithfully embed the symmetric group S_n and permute vectors and matrices they are multiplied against. They are also the extremal points of the Birkhoff polytope B_n of doubly stochastic (n x n) matrices. These are the matrices with non-negative entries whose columns and rows add to 1. The identity matrix I belongs to B_n, and if we let J by the (n x n) matrix whose entries are all 1s, we see 1/n*J is also in B_n. While I is on the exterior of B_n, 1/n*J can be thought of as the center of B_n, in a way that we could make precise. Now we give p_n the uniform probability measure by assigning probability mass 1/n! to each permutation matrix. Let A be a random matrix over p_n. We have the following: E(A) = 1/n*J What this says is that the average value of a random permutation matrix is simply the central point of the Birkhoff polytope :)

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