a straight-line solution
Most trajectories of a system curve and wander through the plane. But a few special ones travel in a perfectly straight line, never turning, only sliding in and out along a fixed ray through the origin. These are the simplest motions the system has, and they are the skeleton on which all the others hang.
A straight-line solution arises from a real eigenvalue lambda with real eigenvector v: x(t) = c e^(lambda t) v. At every instant the position is some scalar multiple of the fixed vector v, so the point stays exactly on the line through the origin in the direction of v — it never leaves that line. The scalar c e^(lambda t) sets how far along it sits and which way it moves: if lambda > 0 the point races outward to infinity along the ray; if lambda < 0 it slides inward toward the origin; if lambda = 0 it sits frozen at a fixed point. The line itself is invariant: start on it and you stay on it forever.
Straight-line solutions matter because they organize the entire phase portrait. The eigenvector directions are the few special lines along which motion is pure exponential, and every other trajectory bends in relation to them — pulled toward the decaying directions, flung out along the growing ones. Reading off the straight-line solutions first is the quickest way to sketch what a 2-by-2 linear system does.
For A = [3, 0; 0, -1], the eigenvector (1, 0) with lambda = 3 gives x(t) = c e^(3t)(1, 0) — motion racing outward along the x1-axis. The eigenvector (0, 1) with lambda = -1 gives motion sliding inward along the x2-axis.
On an eigenvector line the motion is one-dimensional: pure growth or decay, never leaving the line.
Straight-line solutions exist only along REAL eigenvector directions. A system with only complex eigenvalues (a center or spiral) has no straight-line solutions at all — every trajectory curves.