Matrices as Motions
Watch a matrix act on the unit circle: rotation, scaling, shear, and the collapse that makes a regression unsolvable.
The matrix
At t = 0 nothing moves; at t = 1 the full matrix is applied.
Straight lines stay straight and parallel — that is linearity.
Directions the matrix only stretches.
Where the most and least stretching happens.
The unit circle becomes an ellipse
Every linear map in the plane is a rotation, an axis-aligned stretch, and another rotation.
Eigenvalues
Real eigenvalues exist only when the discriminant is non-negative.
Singular values
Always real, always non-negative, always defined.
The product of the singular values is the absolute determinant — the area of the ellipse divided by π.
Reading a matrix as four unrelated numbers hides everything that matters. The same four numbers say how area scales (the determinant), which directions survive unrotated (the eigenvectors), and how badly the map compresses one direction relative to another (the condition number) — the quantity that governs regression stability in Chapter 3.