The SVD is rotate, scale, rotate

Every matrix factors as A = UΣV^T with U, V orthogonal and Σ diagonal. Geometrically that means any A acts on the plane in three simple steps: V^T rotates, Σ stretches the axes by the singular values σ₁, σ₂, and U rotates again. So A maps the unit circle to an ellipse whose half-axes are σ₁, σ₂. Edit A, then press Play to watch the three steps, or step through them.