Simultaneously unitarily diagonalizeable matrices commute

[bmanbs2]

[tex][/tex]

Homework Statement

Matrices A and B are simultaneously unitarily diagonalizeable. Prove that they commute.

Homework Equations

As A and B are simultaneously unitarily diagonalizeable, there exists a unitary matrix P such that

$$P^{-1}AP = D_{1}$$ and $$P^{-1}BP = D_{2}$$, where D$$_{1}$$ and D$$_{2}$$ are diagonal matrices., where D$$_{1}$$ and D$$_{2}$$ are diagonal matrices.

Diagonal matrices always commute.

A unitary matrix multiplied by its conjugate transpose results in an identity matrix.

The Attempt at a Solution

So far I have that
$$P^{-1}APP^{-1}BP = P^{-1}ABP = D_{1}D_{2}$$ and
$$P^{-1}BPP^{-1}AP = P^{-1}BAP = D_{2}D_{1}$$
$$D_{1}D_{2} = D_{2}D_{1} P^{-1}ABP = P^{-1}BAP$$.

Is this enough to show that AB = BA? Where would I use the fact that P is unitary?

Also, how do I delete the latex part at the top? It goes straight to the first latex entry. I put empty tex tags at the top to make it as non-distracting as possible, but if I don't add them, the gibberish goes straight to the next latex section.

 

[bmanbs2]

Bump?

 

[wisvuze]

you don't need the unitary part, since two diagonalizable matrices are generally simutaneously diagonalizable if and only if they commute; the unitary part is needed if you also assert that the matrices are both normal -- did you leave that out?

 

[bmanbs2]

I didn't, but I just recieven an update to the assignment that says I need to show the matrices are normal.