On Oct 15, 2009, at 6:24 PM, 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.
Does anyone know of an algorithm to do this?
I surely don't. My linear algebra is a distant and not entirely
pleasant memory. I went searching and ended up at Google books reading
a couple of texts on real analysis and stochastic matrices. Both
citations reminded me that a matrix induces or implies a polynomial
form over powers of <some> matrix (? constructed out of the basis
vectors?) with the eigenvalues representing the coefficients. I
believe that form can be solved for, but I don't know the process. It
made me wonder if constructing a matrix out of powers of an
appropriate stochastic basis would deliver the sought for resultant.
Such a construction is outside my abilities.
--
David Winsemius, MD
Heritage Laboratories
West Hartford, CT
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