Eigenvectors in this visualization
A 2 × 2 matrix is a linear transformation of the plane, and it can turn a vector off its own line. An eigenvector is one it does not turn: Av = λv, where the eigenvalue λ says how much v is stretched, shrunk or reversed. Every nonzero multiple of an eigenvector is another one with the same λ, so what matters is the line.
The picture follows every direction at once. The dashed circle is the unit circle, the ellipse is its image under A, and each short needle on the circle points the way A sends that point. With the default matrix, turn the red probe v and watch the yellow arrow Av: it leans to one side of v’s line, then to the other. Only on the blue lines does Av lie along v. Those are the eigen lines, each labelled with its λ.
Why are the eigenvalues the roots of det(A − λI)? Rewrite Av = λv as (A − λI)v = 0. A nonzero v can satisfy this only if A − λI squashes the plane flat and sends a whole line to the origin, and that happens exactly when its determinant is 0. The A − λI window draws p(λ) = det(A − λI) as a curve. Drag λ onto a root and the image of the unit square collapses onto a line, or onto the origin if A = λI. The nonzero vectors sent to the origin are the eigenvectors for that λ.
Two independent eigenvectors e₁ and e₂ make a basis in which A only stretches, λ₁ times along e₁ and λ₂ times along e₂, so in these coordinates its matrix is diagonal. Eigenbasis draws one cell of this basis and its image, with corner labels counted in e₁ and e₂, not in x and y. Not every matrix has such a basis. p(λ) is a quadratic, and it can have no real root, as for Turn 90°, which turns every direction. It can also have a double root, as for Shear, which keeps only one line.
Apply repeatedly multiplies v by A again and again and marks the directions of v, Av, A²v and so on. When the eigenvalues are real and differ in size, each step multiplies v’s two eigen components by them, so the component whose eigenvalue is larger in size wins. The directions settle onto its line, unless v starts on the other one. The power method finds an eigenvector this way. For Swap, whose eigenvalues are 1 and −1, a vector off both lines flips between two directions forever.
Things to notice
- The projection v·Av in the readout does not by itself tell whether v is an eigenvector. Av is a multiple of v only on an eigen line, and only there does the page write Av = λv.
- The points of a run lie on the circle because only directions are drawn. The vectors themselves can grow or shrink, and during a run the status line gives each step’s length factor.
- When λ is a root and A ≠ λI, the yellow dashed line is where A − λI sends the plane. λ’s own eigen line, the one sent to the origin, is the heavier blue one. For a double root the two can be the same line.
Controls
- Where v and a column tip overlap, a press takes v. Click once without dragging and the next press takes the other.
- In the A − λI window, press anywhere on the axis to move λ there. λ is caught on the roots and on whole numbers.