Symmetric Matrix as Rotate–Scale–Rotate
Fundamentals of AI/ML · Prof. Parikshit Pareek & Prof. Jitin Singla · IIT Roorkee
A symmetric matrix A admits the decomposition A = U Λ U⊤, where U is an orthogonal matrix (rotation) and Λ is diagonal. Geometrically, applying A to any vector is a three-step recipe: rotate into the eigen-basis, scale by the eigenvalues, rotate back. Drag the timeline to watch the unit circle deform.
A = U Λ U⊤, U = R(θ), Λ = diag(λ1, λ2)
0 · input x
1 · U⊤x
2 · ΛU⊤x
3 · UΛU⊤x = Ax
Stage 0: Input
We start with the unit circle and the eigen-directions u1, u2 tilted by angle θ from the standard axes. The green dot is a sample point we'll trace through the transformation.
current shape
eigen-directions
tracked point (drag at t=0)
original circle
Cumulative matrix at this t
At t = 0: identity. At t = 3: this is exactly A.
The matrix A = U Λ U⊤
Try setting λ2 = λ1: A becomes a pure scaling (any direction is an eigenvector).
What to notice during class
- Between t=0 and t=1, the circle is unchanged — rotations preserve circles. Only the eigen-arrows move.
- Between t=1 and t=2, the axis-aligned scaling Λ turns the circle into an axis-aligned ellipse with semi-axes λ1, λ2.
- Between t=2 and t=3, the rotation U tilts the ellipse so its principal axes point along the original eigen-directions.
- Set λ2 very small (e.g. 0.2): the ellipse collapses toward a line. This is what happens in PCA when one direction has near-zero variance.