Prove that A and B are simultaneously diagonalizable.

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SUMMARY

The discussion centers on proving that two commuting diagonalizable matrices, A and B, are simultaneously diagonalizable. The key argument is based on the property that if AB = BA, then there exists a unique matrix C such that CBC-1 equals the Jordan form J(C) of C. The proof concludes that since C is unique, A must equal the identity matrix I, confirming that both A and B can be diagonalized simultaneously.

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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.
 
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Can you assume A has an inverse?
 
I figured it out.
 

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