(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?

**Physics Forums | Science Articles, Homework Help, Discussion**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Prove that two commuting, diagonalizable operators are simultaneously diagonalizable.

**Physics Forums | Science Articles, Homework Help, Discussion**