eigen spaces of similar matrices

Question

Suppose $A,B $ are similar matrices.Suppose $\alpha $ is an eigen value of $A$.Will the eigen space of $A$ with respect to $\alpha$ be the same as that of the eigen space of $B$ with respect to $\alpha$?If not will they have the same dimension?

Answer 1

Suppose $B = PAP^{-1}$, and let $E$ denote the eigenspace of $A$ associated with some $\alpha$. We define $$ P E = \{P x : x \in E\} $$ Note that for all $y = Px \in PE$, we have $$ By = PAP^{-1}(Px) = P(Ax) = \alpha Px = \alpha y $$

Answer 2

The answer is yes. Similar matrices are representation of the same linear operator in different bases. Eigenspaces corresponding to an eigenvalue are essentially (geometrically) the same but the eigenvector(s) have different coordinates in the new basis, e.g., $y=Px$ as in Omnomnomnom's answer.