 Solution

ID: past-exam-of-the-year-1-of-the-computer-science-course-of-the-university-of-oxford/2019/digital-systems-linear-algebra-introduction-to-formal-proof-resit/6/a/i/solution

det (A - λ I) = (2 - λ) (10 - λ) - 9 = λ^{2} - 12 λ + 11 = (λ - 1) (λ - 11) .

Thus the eigenvalues are 1 and 11. Corresponding eigenvectors are

(3, - 1)^{T}

and

(1, 3)^{T}

. Their dot product is zero and each has Euclidean norm

10

. Normalizing gives

P = \frac{1}{10} (3 - 1 13), D = (10011) .

The columns are orthonormal, so

P^{T} P = I

A P = P D

and hence

P^{T} A P = D

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