Answer the following questions for the following stochastic matrix [P]:
[P] =
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1/2
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1/2
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0
|
|
0
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1/2
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1/2
|
|
0
|
0
|
1
|
(a) Find [Pn] in closed form for arbitrary n > 1.
(b) Find all distinct eigenvalues and the multiplicity of each distinct eigenvalue for [P].
(c) Find a right eigenvector for each distinct eigenvalue, and show that the eigenvalue of multiplicity 2 does not have two linearly independent eigenvectors.
(d) Use (c) to show that there is no diagonal matrix [η] and no invertible matrix [U] for which [P][U] = [U][η].
(e) Rederive the result of (d) using the result of (a) rather than (c).
Text Book: Stochastic Processes: Theory for Applications By Robert G. Gallager.