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stochastic matrix

for a stochastic matrix (each column entries sum to 1), how do I:

a) show that one of the eigenvalues is 1

b) show that an n x n matrix also has an eigenvalue of 1

I can do it for a specific case, not for a general case. some help?

Reply 1

generalebriety

Take such a matrix M; if you can show that det(M - I) = 0, you're done (why?). In order to do that, notice that each column vector of (M-I) is an n-dimensional vector with entries adding to 0; but then notice that these must be linearly independent (e.g. show that the vector space V of n-dimensional column vectors with entries adding to 0 is (n-1)-dimensional).

Reply 2

Zhen Lin

More rapidly, you can show that the row vector

\begin{pmatrix} 1 & \ldots & 1 \end{pmatrix}

is a row-eigenvector, with eigenvalue 1. But then any eigenvalue corresponding to a row-eigenvector is an eigenvalue corresponding to a column-eigenvector (since