Past exam of the algebra 1 and differential equations 1 course of the University of Oxford / 2015 / 2 / b / v

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- Solution 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

is the change-of-basis matrix, then

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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