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