If the vectors $x_1$ and $x_2$ are in the columns of $X$, then what are the eigenvalues and eigenvectors of $$B = XAX^{-1},$$ where $$A= \begin{bmatrix}2&3\\0&1\end{bmatrix}$$
Is there any way to find the eigenvalues and eigenvectors easier than multiplying the whole matrix and then solving $\det(B-\lambda I) = 0$?
The matrices $A$ and $XAX^{-1}$ are similar and therefore they have the same eigenvalues: $1$ and $2$.
On the other hand, if $v$ is an eigenvector of $A$, then $Xv$ is an eigenvector of $XAX^{-1}$. So, compute the two eigenvectors $v_1$ and $v_2$ of $A$, and then compute $Xv_1$ and $Xv_2$.