A matrix is an instruction for moving the whole plane. Type in a matrix and a vector, then see what the matrix does to it — and hunt for the vectors that stay on their own line.
vector vA v (what the matrix does to v)î lands hereĵ lands hereeigenvector lines
Tip: click or drag anywhere on the graph to move the tip of v.
1. The matrix A
A =
2. The vector v
v =
45°
3. What happened to v?
4. Facts about A
How to use this to explain it
A vector is an arrow from the origin. v = (x, y) means “go x across, then y up”.
A matrix moves every point of the plane at once. Its columns say where î = (1, 0) and ĵ = (0, 1) end up, and everything else follows. The shaded square shows how areas change (the area scale factor is the determinant).
Most vectors get knocked off their line: they are turned and stretched.
An eigenvector is a special vector that stays on its own line. It is only stretched or squashed (or flipped). The scale factor is the eigenvalue λ, so A v = λ v.
Try it: choose the exam matrix, press Play, then drag v around until its labels sit on top of each other. That direction is an eigenvector. Check the number against the eigenvalue in box 4.
Rotate 90° has no real eigenvectors, because every vector gets turned. Spot that by dragging v and noticing it never lines up.