Every point p = (x, y) of the plane is mapped by the matrix M to
M·p. The grid dots and the two coloured basis arrows show exactly
where the standard basis vectors i=(1,0) and j=(0,1) land — those
two landing points ARE the columns of M.
M = [a b; c d]
M·(x,y) = (ax+by, cx+dy)
det(M) = ad − bc
λ = (tr ± √(tr² − 4·det)) / 2, tr = a+d
- Determinant — the signed area scale factor; the unit square (area 1) becomes a parallelogram of area |det(M)|. Negative det flips orientation.
- Eigenvectors — directions that M stretches without rotating; drawn as bright rays only when the eigenvalues are real (discriminant ≥ 0).
- Unit circle → ellipse — the image of the unit circle under M is always an ellipse whose semi-axes are the singular values of M (the core geometric idea behind the SVD).