Lab · c09.s01.L1

Eigen-direction explorer

A matrix leaves a direction alone exactly when that direction is an eigenvector, and the stretch factor on that line is the eigenvalue.

Question: Which directions does a matrix leave alone?

Re-simulated Re-simulated: probe directions from 0 to 315 degrees are drawn against the two matrices shown. Gold dashed lines mark the eigen-directions. Only on those lines do the two arrows become collinear, and the stretch there is the eigenvalue. The angles, stretches and turn readings are computed by data/scripts/c09.s01.L1.py from the two matrices shown.

Lab controls

MatrixSection matrix A

Arrow keys move between the two matrices; Home selects the first, End the last

Probe direction0 degrees

Arrow keys step 45 degrees; Home = 0, End = 315

Frame 1 of 1: 0°: on the λ = 2.0 eigen-lineAt 0 degrees the probe v and its image Av lie on one gold line; the turn is 0 degrees and the stretch is 2.000, the eigenvalue 2.0. The image points along v.

Keyboard: Space plays or pauses, Left and Right step, Home goes to the first frame, End goes to the last frame.

Text description of this lab

A navy arrow v points from the origin along the chosen probe direction; a russet arrow Av shows where the selected matrix sends it. Two gold dashed lines mark the eigen-directions. Off them Av turns away from v; a readout gives turn and stretch. On a gold line the arrows are collinear, along or opposite, the turn reads 0 or 180 degrees, and the stretch equals that eigenvalue. A strip below plots turn against angle, marking the current probe.

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Caption status: re-simulated. Every lab declares one of published, re-simulated, schematic, synthetic or interpolated.