[Math] Is it true that 2 matrices are similar **if and only if ** they have the same Jordan form

Question

Is it true that 2 matrices are similar if and only if they have the same Jordan form?
I know that one direction is correct: if have the same Jordan form -> similar.
Is the other direction correct – similar -> same Jordan form?

Answer 1

If $A$ and $B$ are similar, say$$ A = QBQ^{-1} $$

The Jordan form of $B$ is $$ B = PJ_BP^{-1}\\ A = QPJ_BP^{-1}Q^{-1} = (QP)J_B(QP)^{-1} $$so $A$ and $B$ have the same Jordan form.