Prerequisites: Matrix visualization (ch165), linear transformations (ch164) You will learn:
Diagonal matrices as scaling transformations
Non-uniform scaling and its effect on shapes
Scaling in arbitrary directions (via eigenvectors)
How scaling relates to covariance and whitening
Environment: Python 3.x, numpy, matplotlib
# --- Scaling via Matrices ---
import numpy as np
import matplotlib.pyplot as plt
plt.style.use('seaborn-v0_8-whitegrid')
def scale2d(sx, sy):
"""Scale x by sx, y by sy."""
return np.array([[sx,0],[0,sy]])
# Uniform and non-uniform scaling
t = np.linspace(0, 2*np.pi, 300)
circle = np.row_stack([np.cos(t), np.sin(t)])
fig, axes = plt.subplots(1, 4, figsize=(16,4))
configs = [
(scale2d(1,1), 'Identity (sx=1, sy=1)'),
(scale2d(2,2), 'Uniform scale (sx=sy=2)'),
(scale2d(2,0.5), 'Non-uniform (sx=2, sy=0.5)'),
(scale2d(-1,1), 'Reflection + scale'),
]
for ax, (S, title) in zip(axes, configs):
result = S @ circle
ax.plot(circle[0], circle[1], 'gray', lw=1, linestyle='--', alpha=0.5, label='Original')
ax.fill(result[0], result[1], alpha=0.3, color='steelblue')
ax.plot(result[0], result[1], 'steelblue', lw=2)
ax.set_title(f'{title}\ndet={np.linalg.det(S):.1f}', fontsize=9)
ax.set_xlim(-2.5,2.5); ax.set_ylim(-2.5,2.5); ax.set_aspect('equal')
ax.axhline(0,color='k',lw=0.4); ax.axvline(0,color='k',lw=0.4)
plt.suptitle('Diagonal (Scaling) Matrices', fontsize=12)
plt.tight_layout(); plt.show()
# Scaling in arbitrary direction (not axis-aligned)
# Stretch by 3 in direction [1,1]/sqrt(2)
v = np.array([1,1]) / np.sqrt(2) # unit direction
stretch = 3.0
# Outer product construction: scale by (stretch-1) in direction v
S_arb = np.eye(2) + (stretch - 1) * np.outer(v, v)
print(f"Arbitrary-direction scale matrix:\n{np.round(S_arb,4)}")
print(f"Scale in direction v={v}: {np.linalg.norm(S_arb @ v):.4f} (expected {stretch})")
print(f"Scale in direction perp: {np.linalg.norm(S_arb @ np.array([-v[1],v[0]])):.4f} (expected 1)")C:\Users\user\AppData\Local\Temp\ipykernel_1312\107915914.py:12: DeprecationWarning: `row_stack` alias is deprecated. Use `np.vstack` directly.
circle = np.row_stack([np.cos(t), np.sin(t)])
Arbitrary-direction scale matrix:
[[2. 1.]
[1. 2.]]
Scale in direction v=[0.70710678 0.70710678]: 3.0000 (expected 3.0)
Scale in direction perp: 1.0000 (expected 1)
4. Mathematical Formulation¶
Diagonal scaling matrix D = diag(d₁, d₂, ..., dₙ):
D[i,j] = dᵢ if i==j, else 0
(Dv)[i] = dᵢ * v[i] — scale each coordinate independently
Properties:
det(D) = ∏ dᵢ — product of diagonal entries
D⁻¹ = diag(1/d₁,...,1/dₙ) (requires all dᵢ ≠ 0)
D is symmetric — scaling commutes
Scaling in arbitrary direction u (unit vector), scale factor s:
S = I + (s-1) * uuᵀ — adds (s-1) times the projection onto u
Eigenvalues: s (in direction u), 1 (perpendicular)7. Exercises¶
Easy 1. What is the inverse of diag(2, 3, 5)? Verify numerically.
Easy 2. Write scale3d(sx, sy, sz) and apply it to a cube. Plot before and after.
Medium 1. Implement whitening(X) that scales data X so that each feature has unit variance. This is diag(std)⁻¹ @ X. Verify the output has std=1 per feature.
Medium 2. Show that any symmetric positive definite matrix A can be written as Q D Qᵀ where D is diagonal (all eigenvalues) and Q is orthogonal. This is the eigendecomposition — scale in eigen-directions.
Hard. Implement anisotropic Gaussian sampling using a scaling matrix: given a covariance matrix Σ, generate samples from N(0, Σ) using L @ z where z ~ N(0,I) and L is the Cholesky factor of Σ.
9. Chapter Summary & Connections¶
Diagonal matrices scale each axis independently.
det(D) = ∏ dᵢ.Scaling in an arbitrary direction u:
S = I + (s-1)uuᵀ.The eigendecomposition
A = QDQᵀreveals the natural scaling axes of a symmetric matrix.
Forward connections:
In ch172 (Diagonalization), every diagonalizable matrix is a rotation-scale-rotation composition
A = PDP⁻¹.In ch173 (SVD), the singular values in Σ are the scaling factors of the transformation.