Past exam of the year 1 of the computer science course of the University of Oxford / 2022 / Digital systems, linear algebra, introduction to formal proof resit / 4 / c

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- Solution c

Solution

The characteristic polynomial is

det (t I - A) = det (t - 1 - 1 - 1 t) = t^{2} - t - 1.

Its two distinct real roots are the eigenvalues

φ = \frac{1 + 5}{2} > 0, ψ = \frac{1 - 5}{2} < 0.

Their product is

- 1

, so

ψ = - 1/ φ

. Equivalently,

φ^{2} = φ + 1

implies

1/ φ = φ - 1

. The positive eigenvalue is the golden ratio.

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