Eigenvectors — the series

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.

part 1 · algebra

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.

part 2 · algebra

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.

part 3 · algebra

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.

part 4 · algebra

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.

part 5 · algebra

Discs from the rows, and the eigenvalues inside them. The complex plane with one disc per row of a 3×3 matrix — centred on its diagonal entry, with radius the sum of the sizes of the rest of the row — and the 3 eigenvalues marked: 4.14, −5.09, −1.06.

Discs that fence in the eigenvalues

Draw one disc for each row of a square matrix, centred on the diagonal entry, with a radius equal to the sum of the sizes of everything else in that row. Every eigenvalue lies inside one of the discs, and a group of discs set apart from the rest holds exactly as many eigenvalues as it has discs. Nothing is solved to find them.

part 6 · algebra

The largest value on each plane, over every direction of space. A longitude–latitude map of directions u in three dimensions, each shaded by the largest value of xᵀAx on the plane perpendicular to u. The smallest such value is the middle eigenvalue 1.829, reached at ±v₁; the largest is 3.622, reached along a great circle.

The highest point on the sphere is an eigenvalue

For a symmetric matrix, walk a unit arrow over every direction and record the value of xᵀAx. The highest value reached is the largest eigenvalue, the lowest is the smallest, and every eigenvalue in between is a saddle height, a minimum of maxima. From that one description comes a theorem no formula for the roots could give: delete a row and its column, and every eigenvalue of what is left sits between two of the original's.

part 7 · algebra

Eigenvalues that will not crowd together. Unfolded spacings, eigenvalues: 0.36 1.52 1.15 0.88 1.23 1.13 0.34 0.75 2.05 0.65 1.66 0.64 1.07 1.01 0.86 1.23 0.69 1.30 0.77 0.94 0.99 0.71 0.92 2.49 0.32 1.07 0.62 1.34 0.07 0.51 0.98 2.25 0.64 0.78 0.35 0.95 1.36 1.89 0.21 0.79 0.46 0.84 1.12 1.52 0.96 0.65 1.09 0.29 1.30 1.06 1.83 1.50 0.28 0.43 1.19 1.01 0.52 1.54 1.51 1.56. Independent points: 0.34 0.01 0.31 3.72 0.25 2.29 0.98 2.14 1.99 0.54 0.15 0.58 0.96 0.22 0.26 1.11 0.04 0.38 0.55 0.74 0.89 0.11 1.47 0.52 0.06 2.11 1.29 3.04 0.44 0.28 2.49 3.97 0.87 1.04 0.83 0.77 0.79 0.69 1.76 1.82 0.14 0.07 1.03 0.88 1.05 0.37 0.06 0.82 0.70 1.72 0.90 1.24 0.32 0.25 1.07 0.55 1.59 3.23 0.36 0.40. Gaps under a quarter of the mean: 2 against 9.

Eigenvalues that keep their distance

Fill a large symmetric matrix with independent random numbers and its eigenvalues do something no list of independent numbers does: they refuse to crowd together. The reason is a count of conditions in a two-by-two matrix, and its consequences — a semicircle, gaps that vanish at nought, a spectrum almost as rigid as a lattice — reach from heavy nuclei to the zeros of the zeta function.

part 8 · algebra