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

That is,

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 eigen

*vectors* that determine the straight-line motion. But in order to find the eigenvectors, of course, you have to find the eigenvalues.

Hope this helps.