my thought is that the we can use minimal polynomial to write down the rational canonical form of the matrix and then we can use this canonical form to write down the characteristic polynomial and then we deduce that the minimal polynomial and the characteristic polynomial share the same set of roots
How to show that the minimal polynomial and the characteristic polynomial has the same set of roots?
Asked
Active
Viewed 90 times
0
-
Cf. this question – J. W. Tanner Aug 09 '21 at 03:14
-
If it's only roots (and not irreducible factors), by Cayley-Hamilton it is enough to show every root of the characteristic polynomial is a root of the minimal; take a root $\lambda$, an eigenvector $v$, and extend to a basis. Show that for any polynomial $p$, $p(T)(v) = p(\lambda)v$. – Arturo Magidin Aug 09 '21 at 03:15