SVD: M as rotate, scale, rotate

What you are seeing: the singular-value decomposition of a 2x2 matrix $M = U S V^T$ visualized as four panels left to right: the unit circle, after $V^T$ (a rotation), after $S = \mathrm{diag}(s_1, s_2)$ (axis-aligned scaling), and after $U$ (a rotation). The final panel is the image of the unit circle under $M$ , which is the same ellipse as the right panel of the eigenvector playground, just decomposed.

The singular values $s_1 \ge s_2 \ge 0$ are the lengths of the ellipse semi-axes. Sliding the matrix entries shows that rotations keep both $s_i = 1$ (no stretching), pure scaling preserves the coordinate axes, and shears mix the two rotations into something visually interesting. Symmetric $M$ has $V = U$ (up to sign), so the two rotations match.

Figure 1. SVD of a 2x2 matrix. Method: closed-form via eigenvalues of

M^T M

M = [[a, b], [c, d]]

a1.50

b-0.70

c0.40

d2.10

INTERPRETATIONS

Pure rotation: Set all to cos(theta), -sin(theta), sin(theta), cos(theta). You get s_1 = s_2 = 1 and both panels stay circles.
Pure scaling: Set b=c=0 and vary a, d. Then V^T does nothing (circle stays circle), S stretches it, and U does nothing.
Singular matrix: Set ad = bc to make the determinant zero. The ellipse collapses to a line segment; s_2 → 0 and cond(M) → infinity.
Symmetric matrix: Set b=c. The first and last rotations are the same; V = U (up to sign). The ellipse axes align with the coordinate axes.