Why a simple eigenvalue is regular?

Question

Title says all: I can't understand this basic proof. A simple eigenvalue is an eigenvalue with multiplicity algebraic of 1; but why the m. algebraic is equal to the geomtric one?

Answer 1

For some number $\lambda$ to be an eigenvalue, there must be some eigenvector associated with it: $A - \lambda I$ has determinant $0$ so has non-trivial kernel.

On the other hand, there can't be more than one associated eigenvector because the geometric multiplicity is bounded by the algebraic multiplicity.

Therefore, the geometric multiplicity is at least 1 and is at most 1: it is exactly 1.