(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

andABare commuting diagonalizable linear operators. prove that they are simultaneously diagonalizable.

2. Relevant equations

A=BBA

3. The attempt at a solution

We deal with the problem in the Jordan basis of, whereAis diagonal, as Jordan forms are unique.A

Then by rearranging the basis vectors, we can treatas a block diagonal matrix, where the blocks are of the formAλ._{i}I

I aim to prove that, ifis diagonal, and commutes withA, thenBmust also be diagonal, so they have the same Jordan basis.B

I can prove thatmust also be a block diagonal matrix, with the dimensions of the blocks mirroring those ofB.A

This is because if a nonzero entry exists outside of and of's blocks, the corresponding entries inBandABwould be this entry multiplied by different eigenvectors. So the multiplication would not be commutative.BA

But from here I don't know what to do next. Is there some restriction that a diagonalizable matrix may not be put in block diagonal form where the blocks are not diagonalizable themselves?

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# Homework Help: Prove that two commuting, diagonalizable operators are simultaneously diagonalizable.

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