Show that the column space is the set of vectors x

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(a) Assume throughout that [P] is the transition matrix of a unichain (and thus the eigenvalue 1 has multiplicity 1). Show that a solution to the equation [P]w - w = r - ge exists if and only if r - ge lies in the column space of [P - I], where [I] is the identity matrix.

(b) Show that this column space is the set of vectors x for which π x = 0. Then show that r - ge lies in this column space.

(c) Show that, with the extra constraint that π w = 0, the equation [P]w - w = r - ge has a unique solution.

Text Book: Stochastic Processes: Theory for Applications By Robert G. Gallager.

Reference no: EM13909568

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