Finally, we note that for a defective matrix, with more eigenvalues than independent eigenvectors for eigenvalue 1, the left and right eigenvectors of eigenvalue 1 would be orthogonal. Here the left eigenvector is all 1 except for states that can never be reached, and the right eigenvector is \( p_i > 0 \) except for states that give zero probability. We already require that we can reach all states that contribute to \( p_i \). Therefore the left and right eigenvectors with eigenvalue 1 do not correspond to a defective sector of the matrix and they are unique. The Metropolis algorithm therefore converges exponentially to the desired distribution.