an eigenvector direction
When you watch a linear system flow, almost every direction gets twisted as time passes — a vector pointing one way is dragged to point somewhere else. But a few directions are stubborn: the flow only stretches or shrinks them, never rotates them. Those privileged directions are the eigenvector directions, and they are the natural axes of the system.
An eigenvector direction is the line through the origin spanned by an eigenvector v of A, where A v = lambda v. Along this line, the matrix acts like simple multiplication by the number lambda — there is no mixing with other directions. That is exactly why the line is invariant under the flow of x' = A x: a solution starting on it stays on it, moving as the straight-line solution e^(lambda t) v. The sign of the eigenvalue tells you the behaviour along the direction — outward (lambda > 0), inward (lambda < 0), or frozen (lambda = 0) — and distinct real eigenvalues give a separate eigendirection for each, forming a natural coordinate frame in which the coupled system falls apart into independent one-dimensional pieces.
Eigenvector directions are the scaffolding of the phase plane. They mark the special lines you draw first, they reveal where decoupling by diagonalization is possible, and they distinguish a node (eigendirections pointing the same way in time) from a saddle (one in, one out). A system with complex eigenvalues has no real eigendirections — which is precisely the algebraic reason its trajectories must spiral or circle instead of running along straight lines.
For A = [2, 1; 1, 2], the eigenvectors (1, 1) (lambda = 3) and (1, -1) (lambda = 1) define two eigendirections. On the (1,1) line solutions race outward, on the (1,-1) line they grow more slowly — both outward, so the origin is a node.
Eigendirections are the invariant lines; the flow only stretches along them, never rotates them.
An eigenvector has no preferred length or sign: v and 2v and -v all point along the same eigendirection and give the same invariant line. The direction, not the particular vector, is what is geometrically meaningful.