2020/05/18 by Jess Banks, Jorge Garza-Vargas, Banks, Jess +5
Economics, Econometrics and Finance · Mathematics · #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Numerical Analysis (math.NA) #Probability (math.PR) #Random Matrices and Applications #Spectral Theory (math.SP) #Spectral Theory in Mathematical Physics #Stochastic processes and financial applications
paper · pdf · doi:10.48550/arxiv.2005.08930
openalex publication_date 2020/05/18 · openalex created_date 2020/05/21 · openalex updated_date 2026/07/28
Let Gn be an n × n matrix with real i.i.d. N(0,1/n) entries, let A be a real n × n matrix with \Vert A \Vert ≤ 1, and let γ∈ (0,1). We show that with probability 0.99, A + γGn has all of its eigenvalue condition numbers bounded by O(n5/2/γ3/2) and eigenvector condition number bounded by O(n3 /γ3/2). Furthermore, we show that for any s > 0, the probability that A + γGn has two eigenvalues within distance at most s of each other is O(n4 s1/3/γ5/2). In fact, we show the above statements hold in the more general setting of non-Gaussian perturbations with real, independent, absolutely continuous entries with a finite moment assumption and appropriate normalization. This extends the previous work [Banks et al. 2019] which proved an eigenvector condition number bound of O(n3/2 / γ) for the simpler case of \em complex i.i.d. Gaussian matrix perturbations. The case of real perturbations introduces several challenges stemming from the weaker anticoncentration properties of real vs. complex random variables. A key ingredient in our proof is new lower tail bounds on the small singular values of the complex shifts z-(A+γGn) which recover the tail behavior of the complex Ginibre ensemble when \Im z≠ 0. This yields sharp control on the area of the pseudospectrum Λε(A+γGn) in terms of the pseudospectral parameter ε>0, which is sufficient to bound the overlaps and eigenvector condition number via a limiting argument.