 Solution

ID: past-exam-of-the-algebra-1-and-differential-equations-1-course-of-the-university-of-oxford/2015/2/b/v/solution

Suppose a single basis gives the diagonal matrices

D_{S}, D_{T}

. Entrywise multiplication gives

D_{S} D_{T} = D_{T} D_{S}

. If

P

ST = P D_{S} D_{T} P^{- 1} = P D_{T} D_{S} P^{- 1} = TS .

Thus simultaneous diagonalization implies commutation.

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