5.2 Hunting the Eigenline

The last section was a rigged hunt: four candidate lines were drawn in advance, and you merely had to judge them. Suppose nobody marks the candidates. The honest search is continuous — sweep a line through every direction and watch for the moment the field stops crossing it.

The figure below stages that search in the horizontal plane $z equals 0$ , where our matrix from the last section acts by $(x, y) \mapsto (2 x + y, x + 2 y)$ . A thick amber test line runs through the origin at angle $a$ , controlled by a slider that rotates it from $0$ to $3.14$ — nearly a half-turn, which is all a line needs, since a line at angle $a + π$ is the same line. Two dashed answer lines are hidden in the scene; leave them hidden until step 4.

A cube of small arrows for the vector field F(x) = Ax, colored by length, with a thick amber test line through the origin in the horizontal plane. A slider named a rotates the test line from angle 0 to about pi; at exactly two angles the arrows sitting on the line point along it instead of across it. Two dashed answer lines are hidden until revealed. — The field $F (𝐱) = A 𝐱$ with an amber test line $open parenthesis t cos a, t sin a, 0 close parenthesis$ through the origin, rotated by the slider $a$ . At exactly two angles in the sweep, the arrows sitting on the line point along it instead of across it; dashed answer lines are hidden until you reveal them.

Explore the figure

Start at the default $a equals 0.3$ and zoom toward the arrows sitting on the amber line. They cross it at a visible angle — this direction is being turned, so it is no eigenline. Notice that $a equals 0$ would be the $x$ -axis: the dashed decoy from last section, now just one more failed angle.
Sweep $a$ slowly upward and watch only the arrows on the line. Somewhere near $a approximately equals 0.79$ they swing into perfect alignment, pointing outward along the line and noticeably long. Park there.
Keep sweeping. Alignment breaks, then returns near $a approximately equals 2.36$ — but differently: the arrows again lie along the line, yet they are shorter, matching the position vectors in length. Two alignments in the whole sweep, with two different strengths.
Reveal the two hidden dashed lines, answer: $open parenthesis t, t, 0 close parenthesis$ and answer: $open parenthesis t, minus t, 0 close parenthesis$ . Drive the slider to each and confirm your parked angles were $π / 4$ and $3 π / 4$ . Every other angle in the continuum — infinitely many candidates — lost the audition.

The alignment condition

The test direction at angle $a$ is $𝐝 = (\cos a, \sin a)$ , and the field there is $A 𝐝 = (2 \cos a + \sin a, \cos a + 2 \sin a)$ . Alignment means $A 𝐝$ is parallel to $𝐝$ , and two plane vectors are parallel exactly when their cross-term vanishes:

$\cos a (\cos a + 2 \sin a) - \sin a (2 \cos a + \sin a) = \cos^{2} a - \sin^{2} a = \cos 2 a$

The mixed products cancel, and the entire misalignment of this field collapses to the function $cos 2 a$ . On the slider's range, $cos 2 a equals 0$ exactly at $a = π / 4 \approx 0.79$ and $a = 3 π / 4 \approx 2.36$ — the two places your sweep caught. Eigen-hunting is root-finding: a smooth misalignment function of the candidate direction, whose isolated roots are the eigenlines. That is why the winners are so rare, and it is the same rarity you met when a span collapsed at a single slider value in Chapter 2: special directions are knife-edge events, not neighborhoods.

An original work of XYZ Homework, built around interactive XYZ 3D figures. Its chapter sequence is aligned to Interactive Linear Algebra (Margalit & Rabinoff, Georgia Tech, GNU FDL); this work is original, copies nothing from it, and is not affiliated with or endorsed by its authors. License: CC-BY-NC-SA-4.0.