4 CHARLES BORDENAVE AND DJALIL
CHAFA¨
I
when for all continuous and bounded function f : C R (respectively f : R R),
lim
n→∞
f dηn = f dη.
Example 1.2 (Spectra of non-normal matrices). The eigenvalues depend con-
tinuously on the entries of the matrix. It turns out that for non-normal matrices,
the eigenvalues are more sensitive to perturbations than the singular values. Among
non-normal matrices, we find non-diagonalizable matrices, including nilpotent ma-
trices. Let us recall a striking example taken from
[´]
S and [BS1, Chapter 10]. Let
us consider A, B Mn(R) given by
A =


⎜0
⎜.

⎜.
⎝0
0 1 0 · · · 0
0 1 · · · 0⎟
.
.
.
.
.
.
.
.⎟
.
.⎟
0 0 · · ·
1⎠
0 0 0 · · · 0



and B =







0 1 0 · · · 0
0 0 1 · · ·
0⎟⎟
.
.
.
.
.
.
.
.
.
.⎟
.
.⎟⎟
0 0 0 · · ·
1⎠
κn 0 0 · · · 0

where (κn) is a sequence of positive real numbers. The matrix A is nilpotent, and
B is a perturbation with small norm (and rank one!):
rank(A B) = 1 and A B
2→2
= κn.
We have λ1(A) = · · · = λκn (A) = 0 and thus
μA = δ0.
In contrast,
Bn
= κnI and thus λk(B) =
κn/ne2kπi/n 1
for all 1 k n which gives
μB Uniform{z C : |z| = 1}
as soon as
κn/n 1
1 (this allows κn 0). On the other hand, the identities
AA∗
= diag(1,...,1,0) and
BB∗
=
diag(1,...,1,κn)2
give s1(A) = · · · = sn−1(A) = 1 and s1(B) = · · · = sn−1(B) = 1 and thus
νA δ1 and νB δ1.
This shows the stability of the limiting singular values distribution under additive
perturbation of rank 1 of arbitrary large norm, and the instability of the limiting
eigenvalues distribution under an additive perturbation of rank 1 and small norm.
We must keep in mind the fact that the singular values are related to the
geometry of the matrix rows. We end up this section with a couple of lemmas
relating rows distances and norms of the inverse, which are used in the sequel.
Lemma 1.3 (Rows and operator norm of the inverse). Let A Mn(C) with
rows R1,...,Rn. Define the vector space R−i := span{Rj : j = i}. We have then
n−1/2
min
1≤i≤n
dist(Ri,R−i) sn(A) min
1≤i≤n
dist(Ri,R−i).
Proof of lemma 1.3. The argument is essentially in [RuVe1]. Since A and
A have same singular values, one can consider the columns C1,...,Cn of A instead
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