Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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?

share|cite|improve this question
Yes.${}{}{}{}{}$ – Andrés E. Caicedo Mar 16 '14 at 16:30
Can you explain please? – CnR Mar 16 '14 at 16:30
Can you prove that similar matrices have the same eigenvalues? Not by using determinants to compute the characteristic polynomial, but by arguing directly in terms of eigenvectors and how similarity acts. If you see how to do this, the general result is just an extension of these ideas. – Andrés E. Caicedo Mar 16 '14 at 16:33

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.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.