Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

I just got a quick practice question here that I think should be simple but I can't find a definitive answer.

Let $A$ be a square matrix such that $A^3=A$. What can you say about the eigen values of $A$?

It is multiple choice and all of the answers are combinations of -1, 1, and 0. I'm pretty sure 0 and 1 are possible but I'm not sure how to prove any of them. Thanks in advance.

share|improve this question
2  
Hint: Write $A^3 - A = A(A^2 - I) = A(A+I)(A-I) = 0$. –  Amzoti Mar 26 '13 at 1:44
2  
Let $v$ be an eigenvector with eigenvalue $c$. Then $cv=Av=(A^3)v=(c^3)v$. So what can you say about $c$? –  Relsiark Mar 26 '13 at 1:46
    
Oh ok, thanks to both of you for the help. It makes it pretty straight forward. –  Chance Mar 26 '13 at 1:53
2  
In general, the eigenvalues are roots of the minimal polynomial for $A$. If $A^3=A$, what do we know about the minimal polynomial for $A$? –  Thomas Andrews Mar 26 '13 at 2:09
    
Chance, if you can solve the problem now, write up an answer and post it. Then after a while you can accept your answer. This is encouraged on this website. –  Gerry Myerson Mar 26 '13 at 12:17

1 Answer 1

up vote 2 down vote accepted

Minimal polynomial of matrix $A$ satisfying $A^3=A$ is given by $x^3-x=0$,so the possible eigenvalues will be $0,1,-1$, depending on the given matrix now, we can select it from MCQ.

share|improve this answer
2  
It's not necessarily the minimal polynomial; what you can say is that the minimal polynomial is a factor of $x^3-x$. However, the conclusion about the possible eigenvalues is correct; one can't say more because it's not possible to say what the minimal polynomial or the characteristic polynomial are. –  egreg May 22 '13 at 17:34

Your Answer

 
discard

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.