If you write this system in vector/matrix form, you have
where is as you've defined it, and
The eigenvalues of the matrix are those numbers such that for a nonzero eigenvector That is, if you apply the matrix to an eigenvector, you don't change its direction (except possibly an exact reversal in the case of a negative eigenvalue). Hence, if you have an eigenvector with corresponding eigenvalue then
What this DE is saying, geometrically, is that any change in the vector can only be an elongation or a reversal. I can't have any rotations. That is, it must be straight-line motion.
In summary, I would partially agree/partially disagree with the statement. It's the eigenvectors that determine the straight-line motion. But in order to find the eigenvectors, of course, you have to find the eigenvalues.
Hope this helps.