The Group Inverse Associated with an Irreducible Periodic Nonnegative Matrix

  • Authors:

  • Steve Kirkland

  • Affiliations:

  • -

  • Venue:
  • SIAM Journal on Matrix Analysis and Applications
  • Year:
  • 1995

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Abstract

Suppose that $\bf M$ is an irreducible stochastic matrix with period $d\geq 2$. The canonical form for $\bf M$ that exhibits its periodic structure generates a natural partitioning of $\bf M$, which in turn generates a partitioning of $({\bf I} - {\bf M})^\#$, the group generalized inverse of ${\bf I} - {\bf M}$. We derive a formula for the blocks in the partitioned form of $({\bf I} - {\bf M})^#$, discuss possible sign patterns of $({\bf I} - {\bf M})^\#$, and use the partitioned formula to obtain information about the Markov chain associated with $\bf M}.