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Transforms that Preserve The Dominant Eigenvector?

  1. Jun 30, 2008 #1
    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?
     
  2. jcsd
  3. Jul 2, 2008 #2
    Supposing [itex]A \textbf{v} = \lambda \textbf{v}[/itex], where [itex]\textbf{v}, \lambda[/itex] is the dominant eigenvector/eigenvalue pair with components [itex]v_1, v_2, ..., v_n[/itex]. Then you could do something like
    [tex]B = \lambda \left[\begin{matrix} 1 & 0 & 0 & ... & 0 \\ \frac{v_2}{v_1} & 0 & 0 & ... & 0
    \\ \frac{v_3}{v_1} & 0 & 0 & ... & 0 \\ \vdots & \vdots & \vdots & \ddots & 0 \\ \frac{v_{n-1}}{v_1} & 0 & 0 & ... & 0 \\ 0 & \frac{v_n}{v_2} & 0 & ... & 0 \end{matrix}\right][/tex]

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