Let $A$ be a real $n \times n$ matrix with real eigenvalues, and let $PJP^{-1}=A$ be the Jordan decomposition. I'm interested what the columns of $P$ and $P^{-1}$ are when their associated eigenvalues have multiplicity one.
Suppose the eigenvalue $\lambda_i \in \sigma(A)$ has multiplicity 1. My intuition tells me that the $i$th row of $P$ is an eigenvector of $A$ associated with $\lambda_i$, and the $i$th row of $P^{-1}$ is an eigenvector of $A^T$ (i.e. the right eigenvector of $A$) associated with $\lambda_i$. Is this true? If not, then what exactly are these twos supposed to be? I would appreciate any sources that go into detail on the properties of the columns of $P$ and $P^{-1}$.