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 / 4 / f

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- Solution f

Solution

The

n

independent eigenvectors

v_{1}, \dots, v_{n}

of

B

from part (e) are also eigenvectors of

A

. Put

S = (v_{1} \dots v_{n})

and

P = S^{- 1}

. Then

P A P^{- 1} = diag (μ_{1}, \dots, μ_{n}), PB P^{- 1} = diag (λ_{1}, \dots, λ_{n}) .

This supplies the required simultaneous diagonalization over the same field as in part (e).

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