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Ladder

Eigenvectors — the ladder

5 distinct arguments against one idea, from the one that introduces it to the one that assumes the rest.
  1. The directions the map leaves alone. Unit vectors and their images under the map. On the two marked lines the image points the same way as the original, stretched by 3.00 and 1.00.

    The directions a map leaves alone

    Almost every arrow comes out of a transformation pointing somewhere else. A few come out pointing exactly where they went in, only longer or shorter. Those few decide nearly everything the map does.

    rung 1 · algebra
  2. The polynomial whose roots are the stretches. The determinant of A − λI plotted against λ for the map [2, 1, 1, 2], with its roots at 3 and 1 marked.

    The polynomial whose roots are the stretches

    Finding the directions a map leaves alone means finding the numbers at which it crushes something to nothing. Those numbers are the roots of one quadratic, and everything the map does is written in its two coefficients.

    rung 2 · algebra
  3. The same map, written in the basis of its own eigenvectors. Three panels: the map [2, 1, 1, 2] on the standard grid, the diagonal stretch by 3 and 1 it becomes on the eigenvector grid, and the two put back together.

    The same map in a better basis

    Measured along its own invariant directions, a linear map stops shearing and becomes two independent stretches. Nothing about the map has changed; the grid it is described against has.

    rung 3 · algebra
  4. The level curve, and the axes the matrix chooses. The curve xᵀAx = 1 for the matrix [2, 0.8, 0.8, 1.4], drawn by solving for the radius at each angle, with the two eigen-directions marked; they cross at a right angle and are the axes of the curve.

    Symmetry forces a right angle

    A matrix equal to its own reflection across the diagonal always has real stretches and always has perpendicular directions to stretch along. Neither is true of matrices in general, and both follow from one line of algebra.

    rung 4 · algebra
  5. What the map does to a circle. The unit circle with two perpendicular directions marked, and its image under [1.6, 1.2, −0.4, 1.1] — an ellipse whose axes are the images of those two directions, of lengths 2.04 and 1.10.

    What a map does to a circle

    Every linear map sends the unit circle to an ellipse. Two perpendicular directions go to two perpendicular directions, whatever the map is — even a map with no invariant direction at all, and even one that is not square.

    rung 5 · algebra

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