Prove that A and B are simultaneously diagonalizable.

[Daron]

Homework Statement

A and B are commuting diagonalizable matrices. Prove that they are simultaneously diagonalizable.

Homework Equations

AB = BA

The Attempt at a Solution

I have what looks like a proof, but I'm not very happy with it. Is there anything wrong here?

AB = BA
B = ABA^-1

Every matrix has exactly one jordan form. All diagonal matrices are jordan forms. So there exists a unique marix C such that
CBC^-1 = J(C) where J(C) is C's Jordan form.
J(C) = CBC^-1 = CABA^-1C^-1 = (CA)B(CA)^-1
As C is unique, C = CA, so A = I
And CIC^-1 = I, which is diagonal.
So A and B are simultaneously diagonalisable.

 

[vela]

Can you assume A has an inverse?

 

[Daron]

I figured it out.