Transforms that Preserve The Dominant Eigenvector?

[csguy]

Hi,

I'm working with stochastic matrices (square matrices where each entry is a probability of moving to a different state in a Markov chain) and I am looking for transforms that would preserve the dominant eigenvector (the "stationary distribution" of the chain). What I want to do is to cause the antidiagonal of the matrix to be zero.

I remember studying a host of methods that would preserve the spectrum (e.g. QR method, Jacobi rotation, Householder matrices, etc.), but which methods preserve the dominant eigenvector?

Any suggestions?

 

[maze]

Supposing A \textbf{v} = \lambda \textbf{v}, where \textbf{v}, \lambda is the dominant eigenvector/eigenvalue pair with components v_1, v_2, ..., v_n. Then you could do something like
B = \lambda \left[\begin{matrix} 1 &amp; 0 &amp; 0 &amp; ... &amp; 0 \\ \frac{v_2}{v_1} &amp; 0 &amp; 0 &amp; ... &amp; 0<br /> \\ \frac{v_3}{v_1} &amp; 0 &amp; 0 &amp; ... &amp; 0 \\ \vdots &amp; \vdots &amp; \vdots &amp; \ddots &amp; 0 \\ \frac{v_{n-1}}{v_1} &amp; 0 &amp; 0 &amp; ... &amp; 0 \\ 0 &amp; \frac{v_n}{v_2} &amp; 0 &amp; ... &amp; 0 \end{matrix}\right]

I think B \textbf{v} = \lambda \textbf{v} if you work out the multiplication.