In a homework assignment on ODEs I'm supposed to "calculate $(v^t x)$ along x'=Ax and to interpret the result geometrically", where $v$ is the left eigenvector of $A$, meaning $v^t A= \lambda v^t$.
My question is: What does is mean to calculate $(v^t x)$ "along" something ? We did some theory on systems of linear equations, but we have never calculated "something along something". Could you explain to me how to attack this problem ?
I'm also puzzled by the use of left eigenvectors (In none of the classes I took so far this concept was mentioned): Why does one even bother to define it like this ? An eigenvector is something that belongs to an operator so it seems kind of unnatural for me, to define it as left/right for matrices, for two reasons:
1) for operators there exist only a "right" eigenvectors, $\mathcal{A}(v)=\lambda u $.
2) $v^t A= (A^t v)^t$, so one can always reduce a left eigenvectors to a right one, by transposing the matrix, so defining left eigenvectors seems somehow pointless (please correct me if I'm wrong).