 Solution

ID: past-exam-of-the-year-1-of-the-computer-science-course-of-the-university-of-oxford/2013/digital-systems-linear-algebra-introduction-to-formal-proof/6/e/solution

The factor

(λ - 1)

proves that one is an eigenvalue. To find its normalized eigenvector, solve

P v = v

. The second and third rows give

\frac{1}{4} v_{1} + \frac{1}{2} v_{2} = v_{2}, \frac{1}{4} v_{1} + \frac{1}{2} v_{3} = v_{3},

v_{2} = v_{3} = v_{1} /2

v_{1} + v_{2} + v_{3} = 1

now gives

v = (1/2, 1/4, 1/4)^{T} .

Its entries are nonnegative and sum to one, and direct matrix multiplication verifies

P v = v

. As a probability vector it is the stationary distribution; as resource shares it assigns half to the USA and a quarter to each other region.

 New to topics? Read the docs here!