I would like to get an answer to the following question: Let $\Delta_n$ be the vector space of complex $n$-spinors. A vector $X \in \mathbb{R}^n$ acts on $\Delta_n$ by Clifford multiplication. We can thus for every $X \in \mathbb{R}^n$ define a map $F_{X}: \Delta_n\rightarrow \Delta_n$ given by $F_{X}(\phi)=X \cdot \phi$ for every $\phi \in \Delta_n$, where $\cdot$ denotes the Clifford multiplication. My question now is: Can one determine the eigenvalues and eigenvectors of $F_{X}$. If so, what are they? Every help will be appreciated!