Eigenvectors
Drag the tip of the blue vector v around the origin. The orange vector is its image Av under the matrix A. For most directions Av points somewhere new. Along the green dashed lines, and only there, Av lies on the same line as v, so the matrix has only stretched it (or squashed it, or flipped it). Those lines are the eigen-directions, the stretch factors are the eigenvalues, and the vector snaps to them when you get close. Edit the four entries, or pick a preset, and watch the green lines move or vanish.
Mission. With the opening matrix [[2, 1], [1, 2]], drag v to (1, 1) and read off Av. Then drag it to (1, −1). Before you do, predict both images, and predict what happens to the lengths.
Answer
A(1, 1) = (3, 3): three times longer, same direction, so λ = 3. A(1, −1) = (1, −1): unchanged, so λ = 1. Any other direction is a mix of these two, and the two parts grow by different factors, which is why the image swings off the line. Extension: pick the rotation preset. The green lines disappear and the readout shows complex eigenvalues ±i. A quarter turn leaves no direction alone, and that is exactly what "no real eigenvector" means.
What's going on
A 2×2 matrix sends each vector somewhere, and a good way to see the whole map at once is to watch what it does to the unit circle: the pale ellipse is the image of the pale circle. Most radii of the circle get rotated as well as stretched. An eigenvector is a direction that does not rotate: Av = λv. The scalar λ is its eigenvalue, and λ can be negative (the vector flips) or zero (it is squashed to nothing).
To find them, Av = λv means (A − λI)v = 0, which has a nonzero solution only when det(A − λI) = 0. For a 2×2 matrix that is λ² − (a + d)λ + (ad − bc) = 0: the characteristic equation, with the trace and the determinant as its coefficients. Two real roots give two eigen-directions. A repeated root may give two (the identity leaves every direction alone) or just one (the shear). Complex roots mean there is no real eigenvector at all: the matrix turns everything, like a rotation does. The readout shows all three cases as you change presets.
Why care? If you can write a vector as a combination of eigenvectors, then applying A a thousand times is easy: each piece just gets multiplied by λ¹⁰⁰⁰. That is how the Fibonacci matrix [[1,1],[1,0]] gives the closed formula for Fₙ: its eigenvalues are the golden ratio φ ≈ 1.618 and −1/φ, and Fₙ ≈ φⁿ/√5 because the φ-part takes over. Pick the Fibonacci preset and sweep v round: the image leans toward the φ eigen-direction everywhere.
Sheets: The directions a matrix leaves alone. The PageRank demo is a second eigenvector in action.