Hey, I'm wondering if I have a known set of eigenvalues (-1, +1, 0) for A, if I can prove that the matrix A = A(adsbygoogle = window.adsbygoogle || []).push({}); ^{3}?

I can prove that if A^{3}= A, that the eigenvalues would be −1, +1, and 0. The following is the proof:

A*k=lambda*k

A^{3}*k=lambda^{3}*k

Since A=A^{3}, A*k=A^{3}*k

lambda*k=lambda^{3}*k

lambda*k - lambda^{3}*k = 0

lambda - lambda^{3}= 0

lambda*(1-lambda^{2}) = 0

lambda = 0, -1, +1

Is there any way to prove it the other way around? If I know that the eigenvalues are 0, -1, and +1, can I prove that A^{3}= A?

Thanks!

**Physics Forums | Science Articles, Homework Help, Discussion**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Homework Help: Eigenvalues: Proof that A^3=A

**Physics Forums | Science Articles, Homework Help, Discussion**