 Solution

ID: 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/5/b/iii/solution

They are similar matrices, so their eigenvalues, with algebraic multiplicities, are identical. Explicitly,

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

Moreover their eigenvectors correspond by

A v = λ v ⟺ (D^{- 1} A D) (D^{- 1} v) = λ D^{- 1} v

A

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