Spectral Inverses of Stochastic Matrices

descriptionPublicationkeyboard_double_arrow_right Article 01 Jan 1972 English Publisher:Society for Industrial & Applied Mathematics (SIAM)Journal:SIAM Journal on Applied Mathematics, volume 22, pages 22-26 (issn: 0036-1399, eissn: 1095-712X,

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Authors: Wall, J. R.; Plemmons, R. J.;

doi: 10.1137/0122003

Spectral Inverses of Stochastic Matrices

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Abstract

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.

Keywords

Eigenvalues, singular values, and eigenvectors, Theory of matrix inversion and generalized inverses, Stochastic matrices

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