Which are the possible Jordan normal forms for the stochastic matrices? For some reason I got the idea that they always consist of trivial $1\times 1$ blocks even if eigenvalues of multiplicity $>1$. Is this right, and if so, why?
2026-03-26 06:28:20.1774506500
Possible Jordan decompositions of stochastic matrices
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This is not true. It is equivalent to assuming that the geometric and algebraic multiplicities of each eigenvalue for a stochastic matrix is equal. The counter example is:
$$ A=\begin{pmatrix} 1&0&0\\ 0&\frac 13&\frac 23\\ \frac 23&0&\frac 13 \end{pmatrix}. $$ where the Jordan form is: $$ J=\begin{pmatrix} 1&0&0\\ 0 &\frac 13& 1\\ 0&0&\frac 13 \end{pmatrix}. $$