Diagonalizable matrices with same geometric multiplicity for every eigenvalue similar?

Question

Given $A$ and $B$ diagonalizable matrices similar to a diagonal matrix $C$. That is to say, $A$ and $B$ have the same eigenvalues and for every eigenvalue the geometric multiplicity is the same.

Knowing this, can I determine $A$ and $B$ are similar?

Answer 1

$$A=P^{-1}CP \implies C=PAP^{-1}$$ $$B=Q^{-1}CQ \implies C=QBQ^{-1}$$

$$PAP^{-1}=QBQ^{-1}$$

$$A=P^{-1}QBQ^{-1}P=(Q^{-1}P)^{-1}B(Q^{-1}P)$$

Hence, $A$ and $B$ are similar.

Answer 2

The matrices $A$ and $B$ are similar since both of them are similar to the matrix $C$.