If $ A^3=A $ holds for the real valued matrix $A$, what can we conclude about its eigenvalues?
I thought that if I subtracted from $A$ on both sides, I would get $ A^3-A=0 $ which factors as $ A(A-1)(A+1)=0 $.
Can I then correctly conclude by the Cayley-Hamilton theorem that since the characteristic polynomial would be $ p(x)= x(x-1)(x+1)=0 \rightarrow p(A)=0?$
Would that prove that the eigenvalues of the matrix would be $\alpha_k=0,1,-1?$
Thank you for any input!