Let $F$ be a field. Let $x_1,\ldots,x_k,y_1,\ldots,y_k\in F$ be distinct elements in the field. Consider the $k\times k$ matrix that in position $i$, $j$ has the element $\frac{\prod_{l\neq i}(y_i - x_l)}{\prod_{l\neq j}(x_j - x_l)}$.

What are its eigenvectors (as a function of $x_1,\ldots,x_k,y_1,\ldots,y_k$)?

Let $F$ be a field. Let $x_1,\ldots,x_k,y_1,\ldots,y_k\in F$ be distinct elements in the field. Consider the $k\times k$ matrix that in position $i$, $j$ has the element $\frac{\prod_{l\neq i}(y_i - x_l)}{\prod_{l\neq j}(x_j - x_l)}$.

What are its eigenvectors (as a function of $x_1,\ldots,x_k,y_1,\ldots,y_k$)?

Update: I had a typo in the above expression: the correct expression is $\frac{\prod_{l\neq j}(y_i - x_l)}{\prod_{l\neq j}(x_j - x_l)}$ ($j$ replaces $i$ in the numerator). It's indeed a nice exercise to compute the characteristic polynomial of the matrix I specified above, but the expression for the correct matrix is hairy. If it's known (which would make sense, since the transformation is important), please share!