Spectral Inverses of Stochastic Matrices
James R. Wall, Robert J. Plemmons · SIAM Journal on Applied Mathematics · 1972
It is shown that the nonzero eigenvalues of a stochastic matrix A all lie on the unit circle if and only if $A^2 $ has a stochastic group inverse. This is then used to obtain necessary and sufficient conditions for A to have a stochastic spectral inverse. If A is doubly stochastic with a stochastic spectral inverse $A^s $, then $A^s $ is the Moore–Penrose inverse of A.