Another observation about operator compressions Elizabeth Meckes jointworkwithMarkMeckes CaseWesternReserveUniversity Let M be an n×n Hermitian matrix, and let 1 (cid:28) k ≤ n. Then the empirical spectral distributions of most k ×k principal submatrices of M are about the same. 1.0 ll l l ll 0.8 llll ll 0.6 llll ll 0.4 llll ll 0.2 llll ll ll 0.0 0 2 4 6 8 10 An observation about submatrices by Chatterjee and Ledoux 1.0 ll l l ll 0.8 llll ll 0.6 llll ll 0.4 llll ll 0.2 llll ll ll 0.0 0 2 4 6 8 10 An observation about submatrices by Chatterjee and Ledoux Let M be an n×n Hermitian matrix, and let 1 (cid:28) k ≤ n. Then the empirical spectral distributions of most k ×k principal submatrices of M are about the same. An observation about submatrices by Chatterjee and Ledoux Let M be an n×n Hermitian matrix, and let 1 (cid:28) k ≤ n. Then the empirical spectral distributions of most k ×k principal submatrices of M are about the same. 1.0 ll l l ll 0.8 llll ll 0.6 llll ll 0.4 llll ll 0.2 llll ll ll 0.0 0 2 4 6 8 10 Theorem (Chatterjee-Ledoux) For M given, let A be chosen uniformly at random from all k ×k principal submatrices. Let F denote the empirical distribution A function of A; that is, 1(cid:12) (cid:12) FA(x) = (cid:12){j : λj(A) ≤ x}(cid:12). k Let F(x) := EF (x). Then for r > 0, A √ √ P[(cid:107)F −F(cid:107) ≥ k−1/2+r] ≤ 12 ke−r k/8 A ∞ and √ 13+ 8log(k) E(cid:107)F −F(cid:107) ≤ √ . A ∞ k More formally: For M given, let A be chosen uniformly at random from all k ×k principal submatrices. Let F denote the empirical distribution A function of A; that is, 1(cid:12) (cid:12) FA(x) = (cid:12){j : λj(A) ≤ x}(cid:12). k Let F(x) := EF (x). Then for r > 0, A √ √ P[(cid:107)F −F(cid:107) ≥ k−1/2+r] ≤ 12 ke−r k/8 A ∞ and √ 13+ 8log(k) E(cid:107)F −F(cid:107) ≤ √ . A ∞ k More formally: Theorem (Chatterjee-Ledoux) Let F(x) := EF (x). Then for r > 0, A √ √ P[(cid:107)F −F(cid:107) ≥ k−1/2+r] ≤ 12 ke−r k/8 A ∞ and √ 13+ 8log(k) E(cid:107)F −F(cid:107) ≤ √ . A ∞ k More formally: Theorem (Chatterjee-Ledoux) For M given, let A be chosen uniformly at random from all k ×k principal submatrices. Let F denote the empirical distribution A function of A; that is, 1(cid:12) (cid:12) FA(x) = (cid:12){j : λj(A) ≤ x}(cid:12). k Then for r > 0, √ √ P[(cid:107)F −F(cid:107) ≥ k−1/2+r] ≤ 12 ke−r k/8 A ∞ and √ 13+ 8log(k) E(cid:107)F −F(cid:107) ≤ √ . A ∞ k More formally: Theorem (Chatterjee-Ledoux) For M given, let A be chosen uniformly at random from all k ×k principal submatrices. Let F denote the empirical distribution A function of A; that is, 1(cid:12) (cid:12) FA(x) = (cid:12){j : λj(A) ≤ x}(cid:12). k Let F(x) := EF (x). A and √ 13+ 8log(k) E(cid:107)F −F(cid:107) ≤ √ . A ∞ k More formally: Theorem (Chatterjee-Ledoux) For M given, let A be chosen uniformly at random from all k ×k principal submatrices. Let F denote the empirical distribution A function of A; that is, 1(cid:12) (cid:12) FA(x) = (cid:12){j : λj(A) ≤ x}(cid:12). k Let F(x) := EF (x). Then for r > 0, A √ √ P[(cid:107)F −F(cid:107) ≥ k−1/2+r] ≤ 12 ke−r k/8 A ∞ More formally: Theorem (Chatterjee-Ledoux) For M given, let A be chosen uniformly at random from all k ×k principal submatrices. Let F denote the empirical distribution A function of A; that is, 1(cid:12) (cid:12) FA(x) = (cid:12){j : λj(A) ≤ x}(cid:12). k Let F(x) := EF (x). Then for r > 0, A √ √ P[(cid:107)F −F(cid:107) ≥ k−1/2+r] ≤ 12 ke−r k/8 A ∞ and √ 13+ 8log(k) E(cid:107)F −F(cid:107) ≤ √ . A ∞ k