# can a real matrix have both a real minimal polynomial and complex characteristic polynomial?

I had a test a few days ago, and I had a question there. can there be a real matrix where the minimal polynomial is $$(x^2 - 3x + 2)$$ and the characteristic polynomial is $$(x^2 - 3x + 2)(x^2 - x + 2)$$? I tried solving it for hours, and searched a lot for an answer. also on that note, is it possible for a minimal polynomial to not have all the eigenvalues? such as in this case?

edit:

thank you for the answer, however, how do I prove this without the theorem? can I prove this through the fact that it has 4 distinct eigenvalues, and therefore Diagonalizable?

## 1 Answer

The minimal polynomial and characteristic polynomial must have the same roots. See theorem 1.10 of http://pcwww.liv.ac.uk/~mmertens/MinimalPolynomial.pdf

• so basically, a minimal polynomial will always posses all eigenvalues from the Characteristic polynomial? and that means that it is not possible for there to be a real minimal polynomial and a complex Characteristic polynomial? – bignt Sep 29 '20 at 12:25