I'm trying to compute the first column of $M$ where
$$M=(A - x_1I)(A - x_2I)\cdots(A - x_rI)$$
where $A$ is in $R^{n \times n}$ and $x$ is a vector in $R^r$.
Whatever way I think of it, it requires compute every single column of $A_i=A-x_iI$. There doesn't seem to be anyway around that.
Can anyone suggest a way to compute this?
Does the fact that $A$ is written as $(A - x_1I)$ help in any way?