Eigenvalue Outlier — Spiked Random Matrix

The BBP Transition

Consider a Wishart matrix W = X^T X/n where X is an n×p matrix of iid N(0,1) entries, with an added rank-1 spike: signal strength θ.

W_spiked = W + θ·vv^T

The bulk eigenvalues follow the Marchenko-Pastur law on [λ⁻, λ⁺] where λ± = (1 ± √γ)², γ = p/n.

λ± = (1 ± √γ)² (bulk edges)

The BBP phase transition (2005): an outlier eigenvalue detaches from the bulk only if θ > √γ. Below this threshold, the signal is invisible!

θ > √γ → λ_out = θ + γ/θ + 1
θ ≤ √γ → no outlier (θ invisible)

Signal strength θ: 0.80 Aspect ratio γ = p/n: 0.50 Matrix size n: 120

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