This question is really several short general questions to clear up some confusions. We'll start with where I'm at:
An endomorphism $\phi$ is a map from a vector space $V$ to itself. After choosing a basis $\mathcal{B}$ of $V$, one may determine a matrix to represent such an endomorphism with respect to that basis, say $[\phi]_{\mathcal{B}}$.
Question 1) If given a matrix without a basis specified, could you deduce a unique endomorphism it corresponds to? (my lean is no)
Question 2) In a similarity transform, say $A=SDS^{-1}$ where $D$ is diagonal, $S$ is the change of basis matrix from one basis to another. My question is, since $D$ is diagonal does that mean the matrix $D$ is the same endomorphism as $A$ with respect to the standard basis in $\mathbb{R^{n}}$. Or are we unable to determine which bases are involved if only given the matrices.
Question 3) Given a matrix related to an endomorphism, is it possible to determine the basis used to represent the endomorphism. (Lean yes)
Overarching Question) I am trying to understand what happens under similarity transforms. I understand we input a vector, a change of basis is applied to it, the endomorphism is applied with respect to the new basis, and then it is changed back to the old basis, but my confusion relates to the construction of the similarity matrix. If a matrix is diagonalizable, then $S$ turns out to be the eigenvectors arranged in a prescribed order. Why is this! Why do the eigenvectors for the endomorphism $\phi$ with respect to one basis act as a change of basis matrix, and what basis to they go to? This is really part of question 2. Does this send the vectors to the standard basis? Or some other basis that just happens to diagonalize the endomorphism.
Thanks!