MathOverflow is a question and answer site for professional mathematicians. 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

Let $A$ and $B$ be two $4$ by $4$ matrices. Using Newton's identites, one can prove that if $$det(A) = det(B)$$ and $$tr(A^i) = tr(B^i)$$ for $i=1,2,3$, then $A$ and $B$ have the same characteristic polynomial, thus the same eigenvalues.

I'm interested in pairs of matrices $A$ and $B$ that satisfy all those equations except the last one, i.e. $$det(A)=det(B)$$ $$tr(A)=tr(B)$$ $$tr(A^2)=tr(B^2)$$ but $tr(A^3) \neq tr(B^3)$.

Does anyone know how to generate such matrices? Have they ever been studied? A reference would be nice.

Thank you, Malik

share|cite|improve this question
Isn't that just saying that the characteristic poly's of A and B differ only in the linear term? That gives an easy way to generate such matrices :) – t3suji Jun 17 '10 at 16:27
Yes I was making some additional assumptions unthinkingly t3suji, thanks. Maybe I'll have another go if Malik tells us what kind of properties he/she is interested in. – Q.Q.J. Jun 17 '10 at 20:14
up vote 5 down vote accepted

Such matrices will have a characteristic polynomial $z^4+a_3z^3+a_2z^2+a_1z+a_0$ with the same $a_3$, $a_2$, $a_0$ but distinct $a_1$. You can generate a plenty of diagonal such matrices by picking roots of such two polynomials. I cannot vouch that they were not studied but I am pretty certain that nothing groundbreaking came out of such studies.

share|cite|improve this answer
Thanks, that answers my question as stated. Though, I was also hoping for references that give properties of such matrices. Do you know any? – Kalim Jun 17 '10 at 16:58
Maybe you could be more specific in terms of what kind of properties you're looking for, or why you're interested in these matrices in the first place. The (very restrictive!) properties which define these matrices surely generate more identities than one could possibly know what to do with. – Cam McLeman Jun 17 '10 at 17:06
The question is rather basic to be discussed specifically in a paper. You can refer to any book that does Jordan forms ad elementary symmetric functions, e.g., Artin's Algebra. – Bugs Bunny Jun 17 '10 at 21:18

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.