The stochastic matrix for a damped system is
S = ( 1 − α ) Q + α P = 1 2 Q + 1 2 P = [ 1 / 12 1 / 12 1 / 12 1 / 12 1 / 12 1 / 6 1 / 3 1 / 12 1 / 12 1 / 12 1 / 12 1 / 6 1 / 12 1 / 3 1 / 12 1 / 12 1 / 12 1 / 6 1 / 3 1 / 12 1 / 12 1 / 12 1 / 12 1 / 6 1 / 12 1 / 3 1 / 12 7 / 12 1 / 12 1 / 6 1 / 12 1 / 12 7 / 12 1 / 12 7 / 12 1 / 6 ] {\displaystyle {\begin{aligned}S&=(1-\alpha )Q+\alpha P\\&={\frac {1}{2}}Q+{\frac {1}{2}}P\\&={\begin{bmatrix}1/12&1/12&1/12&1/12&1/12&1/6\\1/3&1/12&1/12&1/12&1/12&1/6\\1/12&1/3&1/12&1/12&1/12&1/6\\1/3&1/12&1/12&1/12&1/12&1/6\\1/12&1/3&1/12&7/12&1/12&1/6\\1/12&1/12&7/12&1/12&7/12&1/6\end{bmatrix}}\end{aligned}}}
When the damping factor α {\displaystyle \displaystyle \alpha } tends to zero, the eigenvalues of S tend to Q. Because Q projects onto a 1 dimension subspace, there is one eigenvalue equal to 1, and the rest are zero.
tends to zero, the eigenvalues of S tend to Q. Because Q projects onto a 1 dimension subspace, there is one eigenvalue equal to 1, and the rest are zero.
