SPD MANIFOLD · VISUALIZATION

What is Positive Definiteness?

2-channel EEG · correlation matrix C = [[1, ρ], [ρ, 1]] · eigenvalues λ₁ = 1+ρ, λ₂ = 1−ρ

POSITIVE DEFINITE — interior of SPD manifold
−1.2
impossible −1
boundary 0
circle +1
boundary +1.2
impossible
↵ ρ = 0. Channels are uncorrelated. Ellipse is a circle. Equal eigenvalues, equal axes. This is the 'center' of the SPD manifold for unit-variance signals.
WHAT POSITIVE DEFINITE MEANS
For every direction v ≠ 0:
vᵀCv > 0
The matrix "pushes back" in every direction.
No direction collapses to zero energy.
Geometrically: the ellipse has real positive area.
EACH EEG WINDOW IS A POINT
One 1s window → one covariance matrix → one ellipse shape
→ one point on the SPD manifold

The manifold is the space of all valid shapes.
Interictal brain = one region. Ictal brain = another.
← Eigenvalues Riemannian Geometry →
COVARIANCE ELLIPSE · data points + 1σ 2σ 3σ contours
QUADRATIC FORM · xᵀC⁻¹x (contours = σ-ellipses)
λ₁ = 1 + ρ = 1.000 axis [1,1]/√2 (+45°)
√λ = 1.000 (half-axis length)
λ₂ = 1 − ρ = 1.000 axis [1,−1]/√2 (−45°)
√λ = 1.000 (half-axis length)