The Metropolis method is simply the power method for computing the right eigenvector of \( \boldsymbol{M} \) with the largest magnitude eigenvalue. By construction, the correct probability distribution is a right eigenvector with eigenvalue 1. Therefore, for the Metropolis method to converge to this result, we must show that \( \boldsymbol{M} \) has only one eigenvalue with this magnitude, and all other eigenvalues are smaller.
Even a defective matrix has at least one left and right eigenvector for each eigenvalue. An example of a defective matrix is
$$ \begin{bmatrix} 0 & 1\\ 0 & 0 \\ \end{bmatrix}, $$with two zero eigenvalues, only one right eigenvector
$$ \begin{bmatrix} 1 \\ 0\\ \end{bmatrix} $$and only one left eigenvector \( (0\ 1) \).