A further idea:
Consider the triangular square matrices of the form with decreasing
eigenvalues on the diagonal:
1 0 0 0 0 0
.3 .7 0 0 0 0
.4 .2 .4 0 0 0
.2 .4 .2 .2 0 0
.1 .3 .4 .2 .1 0
.2 .2 .2 .2 .1 .1
This would have the specified eigenvalue composition. Couldn't you
then apply "Gaussian reduction" row operations not for the purpose of
"reduction" but essentially the inverse? Wouldn't these all be in the
class of matrices that are "row echelon equivalent"? Don't they have
identical eigenvalues? Couldn't you fill in the lower triangular
elements with a random group of elements drawn from runif() calls.
And then a series of Gaussian row operations that preserved the
stochastic character?
e.g. a*Row1 +(1-a)Row6 -> Row1
--
David Winsemius, MD
Heritage Laboratories
West Hartford, CT
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?
Thanks for any help,
Ravi.
----------------------------------------------------------------------------
-------
Ravi Varadhan, Ph.D.
Assistant Professor, The Center on Aging and Health
Division of Geriatric Medicine and Gerontology
Johns Hopkins University
Ph: (410) 502-2619
Fax: (410) 614-9625
Email: rvarad...@jhmi.edu
Webpage:
<http://www.jhsph.edu/agingandhealth/People/Faculty_personal_pages/Varadhan
.
html>
http://www.jhsph.edu/agingandhealth/People/Faculty_personal_pages/Varadhan.h
tml
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David Winsemius, MD
Heritage Laboratories
West Hartford, CT
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