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Why is it, that for the matrix

$A \in \text{Mat}(n\times n, \mathbb{C})$

the characteristic polynomial $\chi_A(t)$ and the minimal polynomial $\mu_A(t)$ have same roots?

Since $\chi_A(t) = \mu_A(t) \cdot p(t)$ it should be easy to follow, that $\chi_A(t)$ has roots where $\mu_A(t)$ has roots.

But why can't $\chi_A(t)$ have roots where $\mu_A(t)$ hasn't?

  • 2
    Actually, in general, the characteristic polynomial and the minimal polynomial have the same irreducible factors.2012-01-22

2 Answers 2