 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/4/c/solution

With the two diagonal matrices from the preceding part, write

D_{A} = P A P^{- 1}

and

D_{B} = PB P^{- 1}

. Their diagonal products agree entry by entry, so

D_{A} D_{B} = D_{B} D_{A}

. Therefore

A B = P^{- 1} D_{A} D_{B} P = P^{- 1} D_{B} D_{A} P = B A .

A, B

are commuting operators.

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