Another observation about operator compressions

75 Pages·2010·0.51 MB·English

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