 Solution

ID: past-exam-of-the-year-1-of-the-computer-science-course-of-the-university-of-oxford/2015/digital-systems-linear-algebra-introduction-to-formal-proof/1/e/solution

λ = 1

is an eigenvalue. To find its normalized eigenvector, solve

(P - I) v = 0

together with

v_{C} + v_{H} + v_{B} = 1

. The first two independent linear equations are

- 4 v_{C} + v_{H} + v_{B} = 0, v_{C} - 2 v_{H} + 2 v_{B} = 0.

5 v_{C} = 1

. The second gives

v_{H} - v_{B} = v_{C} /2 = 1/10

, while

v_{H} + v_{B} = 4/5

. Consequently,

v = 1/5 9/20 7/20, P v = v .

All entries are positive and sum to one. This probability vector is the stationary distribution.

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