From Standard Noise to a General Gaussian

One fixed cloud travels through the affine map X = μ + QΛ^1/2Z. The four panels isolate each operation; the second section follows what the same map does to area and density.

Scale Λ^1/2

√λ_x 1.60

√λ_y 0.75

Rotate Q

Angle θ 30°

Shift μ

μ_x 1.20

μ_y -0.80

What stays fixed

The seeded standard-normal sample is identical in all four panels. Only the displayed transformation changes.

1. Build the affine transformation

Matrix products act from right to left: scale first with Λ^1/2, rotate with the orthogonal matrix Q, then add μ. Because Z ~ N(0, I) has no preferred orientation, we can treat its current coordinates as principal-axis coordinates and begin with the scaling step.

1Standard noise

Z ~ N(0, I)

2Scale the axes

Y = Λ^1/2Z

3Rotate

W = QΛ^1/2Z

4Shift the mean

X = μ + QΛ^1/2Z

B = QΛ^1/2

—

Σ = BBᵀ

—

det Q

1.000

|det B| = √λ_x√λ_y

—

2. Why the Jacobian changes density

The highlighted cell is enlarged so we can see it; treat it as a stand-in for the local differential area dA_z. Under B it becomes a parallelogram with area dA_x = |det B| dA_z. Translation moves the grid but does not change its area.