Is $\text{dim}\, G(T_\mathbb C ,\lambda)=\text{dim}\, G(T,\lambda)$ when $\lambda$ is real?
I am studying the theorem 9.23(c) of the S.Axler's Linear Algebra Done Right (3rd Edition) on page 284 of that book. Then suddenly this question appears in my mind. 9.23(c) says that the eigenvalues of $T$ are $precisely$ the real zeros of the characteristic polynomial of $T$. To be precise :
Let $V$ be a real vector space,let $T\in \mathcal L (V)$. Let $T_{\mathbb C} $ be the complexification of $T$. Let $\lambda$ be an eigenvalue of $T$. Does the equation
$\text{dim null}(T-\lambda I)^{dim V}=\text{dim null}(T_\mathbb C-\lambda I)^{dim V}$
hold ?