(a) the quadratic form, swept over the circleThe cloud is 600 draws of N(0, Σ) with Σ = U diag(λ₁, λ₂) U⊤ set by the sliders. The gold oval is the polar plot of v⊤Σv over unit v — the directional variance of the cloud. Drag v or press sweep: the stacked bar shows the decomposition λ₁α₁² + λ₂α₂², which can never exceed λ₁ and reaches it exactly when v lines up with u₁. The chip checks the fact against the data: the sample variance of v⊤X tracks v⊤Σv at every angle.
λ₁
λ₂
θ
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(b) rank one: one direction, and silence elsewhereLeft: the action of uu⊤. Twelve unit test vectors go in; every output u(u⊤v) lands on the line of u. Along u the eigenvalue is ‖u‖²; along the dashed perpendicular, everything is sent to zero. Drag the tip of u and watch both follow. Right: the same matrix as a covariance perturbation. Draws of N(0, I + c uu⊤) start round at c = 0 and bulge only along u as c grows: the axis along u stretches to √(1+c‖u‖²) while the orthogonal axis stays at 1 — the planted-direction mechanism of Lecture 4 in its simplest form.
perturbation c
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