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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?

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# Transforms that Preserve The Dominant Eigenvector?

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