What is Characteristic polynomial?

$\chi_u(X) = \det(X\,\mathrm{id} - u)$ — computed in any basis as $\det(XI_n - A)$, a monic polynomial of degree $n$, invariant under similarity (Theorem 2.17). Its roots in $K$ are exactly the eigenvalues ($\lambda$ eigenvalue $\iff u - \lambda\,\mathrm{id}$ not injective $\iff \chi_u(\lambda) = 0$), and

The algebraic multiplicity $m_\lambda$ of an eigenvalue is its multiplicity as a root of $\chi_u$; the geometric multiplicity is $\dim E_\lambda$, and $1 \leq \dim E_\lambda \leq m_\lambda$.