The minimal polynomial divides to and so, ...
For a 5-dimensional vector space, I have an operator I'm working with T satisfying
(T-1)(T^2 - 4) = 0, but T =/=1 and T^2 - 4 =/= 0.
I'm not sure how to get started on this. I can find them for a matrix, but in this case I have no matrix, only an equation in T. Any push in the right direction would be great, thank you.
Well then this must mean that the minimal polynomial has to be x + 2, as I don't see any other factors dividing the original equation, and a simple check yields
T + 2I = 0 -> (T+2I)(T+2I) = T^2 + 4T + 4I = 0 is not a contradiction, so the only eigenvalue is -2? That feels right but any input would be great.
I don't know where you got that. Fernando Revilla told you that the minimal polynomial must divide but was NOT nor . Since the orginal equation has three distinct first degree factors, it is the minimal polynomial.
(The "I don't know where you got that" was in response to arcketer's last post.)
Oh, I see. I think I was thrown off by the wording. So having the minimal polynomial, it is obvious the corresponding eigenvalues are 1, 2, -2 each with single multiplicity, and this allows me to make some claims about the possible Jordan Canonical forms. Thank you both very much for your help