[R] compare two matrices

[Kehl Dániel]

Dear Members,

I have two estimated transition matrices and I want to compare them.
In fact I want to check the hypothesis if they come from the same process.
I tried to look for some test but all I found was independence test of 
contingency tables.
The following code shows that the usual chi-squared test statistic does 
not follow chisq distribution.

MCRepl <- 5000
khi12 <- rep(0,MCRepl)

for (i in 1:MCRepl){
   n <- 10000

   M1 <- matrix(table(sample(1:9, size = n, replace = TRUE)),3,3)
   M1 <- M1/rowSums(M1)
   M2 <- matrix(table(sample(1:9, size = n, replace = TRUE)),3,3)
   M2 <- M2/rowSums(M2)

   khi12[i] <- sum((M1-M2)^2/M2)

}
plot(density(khi12))

Is there a procedure implemented in R to make this comparison? If not, 
what could be a nice test statistic?

Thank you so much,
Daniel