Understanding Matrix as operators over Vectors and Eigen Vectors

What to do

  1. Click anywhere on the canvas to choose a point p (blue). Drag the blue point to move it.
  2. Edit the matrix T below. The transformed point updates as q = T p (red).
  3. Eigenvector directions (if real) are shown as dashed lines through the origin.
  4. Enable “Show eigenvalues/vectors” to display the computed λ and v.

Matrix T (2×2)

Legend
Blue: selected vector/point p (arrow from origin)
Red: transformed vector/point q = T p (arrow from origin)
Dashed lines: eigenvector directions (if real)

Key math (2×2)

Transformation: q = T p
Eigenvector equation: T v = λ v (v ≠ 0)
Characteristic polynomial: λ² − tr(T) λ + det(T) = 0
where tr(T)=t11+t22, det(T)=t11·t22 − t12·t21
Fixed view: world coordinates are always [-4,4]×[-4,4] (no auto-rescaling).
If eigenvectors are not shown, it usually means the matrix has complex eigenvectors (e.g., rotation-like). Use Random (real eigenvectors) or adjust T until they appear.