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W is a normal stochastic matrix which has non-negative elements and each row sums to 1.

W can be represented by the factorization (a constraint that can be imposed on the particular system):

W = ED

Where E is a symmetric matrix and D is a diagonal matrix.

How can I calculate the eigenvalues and eigenvectors of W?

W will be large and sparse, any advice with regards algorithms would be greatly appreciated.

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If $W$ is large, sparse, and non-symmetric, you probably want to use the Arnoldi method (see the ARPACK library). This only requires that you can compute matrix-vector products with $W$ and a given (algorithm-provided) vector.

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