Cayley-Hamilton theorem

Theorem

1. The minimal polynomial divides the characteristic polynomial.
2. The characteristic polynomial divides some power of the minimal polynomial
3. The characteristic and minimal polynomials have the same roots (not counting multiplicity)

Proof

This is a corollary of this theorem.

1. Follows directly from the theorem

2. Follows from the fact that $q_i|q_K$ for all $1\le i\le K$. Therefore, ${\chi }_T|q_K^K$.

3. The follows from 1 and 2.