 Solution

ID: past-exam-of-the-year-1-of-the-computer-science-course-of-the-university-of-oxford/2021/digital-systems-linear-algebra-introduction-to-formal-proof/4/b/solution

det (λ I - A) = λ^{2} - λ - 1

. Its distinct eigenvalues are

α = (1 + 5) /2

and

β = (1 - 5) /2

. For either eigenvalue

λ

, the vector

(λ, 1)^{⊤}

is an eigenvector, since

A (λ, 1)^{⊤} = (λ + 1, λ)^{⊤} = λ (λ, 1)^{⊤}

Choose

S = (α 1 β 1), D = (α 0 0 β) .

S

α - β = 5 \neq = 0

, so

S^{- 1} = \frac{1}{5} (1 - 1 - β α) .

Its columns being eigenvectors gives

A S = S D

, and left multiplication by

S^{- 1}

proves

S^{- 1} A S = D

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