Jordan Bases Linear Dependence?

Question

Say we have a Jordan form for $A$, called $J$. Two different people come up with two different bases for $J$; person A comes up with $V$ = {$v_1, v_2, ... , v_n$}, and P.B. comes up with $W$ = {$w_1, w_2, ... , w_n$}. How many pairs of vectors in $V$ and $W$ are we sure are scalar multiples of each other?

Answer 1

There does not have to be any such pair. For example, if $A=I$ is the identity map, it's matrix with respect to any basis of the vector space is identity matrix, which is Jordan. And for any basis $$\{e_1,\dots,e_n\}$$ we can find another that contain no vector colinear with $e_1, e_2, \ldots$ or $e_n$, for example $$\{e_1+e_2,e_1-e_2,e_1+e_2+e_3,\ldots,e_1+\cdots + e_n\}.$$