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

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- Solution 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},

so

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

. The normalization

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.

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