Ravi Varadhan wrote:
Hi,
Given a positive integer N, and a real number \lambda such that 0 < \lambda
< 1, I would like to generate an N by N stochastic matrix (a matrix with
all the rows summing to 1), such that it has the second largest eigenvalue
equal to \lambda (Note: the dominant eigenvalue of a stochastic matrix is
1).
I don't care what the other eigenvalues are. The second eigenvalue is
important in that it governs the rate at which the random process given by
the stochastic matrix converges to its stationary distribution.
An idea...
Form a vector e of desired eigenvalues where the first eigenvalue is
1, the second is \lambda, and the remaining eigenvalues are whatever
is convenient (i*\lambda_i/(N-2) where i=0:(N-3) for example if repeated
eigenvalues would be inconvenient). Then form A=diag(e) and modify it
to make the matrix stochastic by replacing the element just below
lambda_i with (1-lambda_i) until you get to the lambda_N, where A[1,N]
would be set to (1-lambda_N).
If \lambda=0.8, then we could form
e <- c(1, 0.8, 0.6, 0.4, 0.2)
then I think your desired matrix A is
A <- diag(e)
A[3,2] <- 1-e[2]
A[4,3] <- 1-e[3]
A[5,4] <- 1-e[4]
A[1,5] <- 1-e[5]
eigen(A)
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