Let and be transition matrices of some finite ( states) homogeneous ergodic Markov chains, and be their stationary distributions. The article is devoted to the condition number , i.e. the constant for which
New upper bounds for are obtained in terms of and its group inverse . E.g.
are eigenvalues of .
Numerical results show that these bounds are approximately two times better than the estimate of C. D. Meyer for [SIAM J. Matrix. Anal. Appl. 15, No. 3, 715–728 (1994; Zbl 0809.65143)].