[R] Generating a stochastic matrix with a specified second dominant eigenvalue

[David Winsemius]

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: rvaradhan at 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