LECTURE NOTES ON THE CIRCULAR LAW 3

Theorem 1.1 (Weyl inequalities). For every A ∈ Mn(C) and 1 ≤ k ≤ n,

(1.1)

k

i=1

|λi(A)| ≤

k

i=1

si(A).

The reversed form

n

i=n−k+1

si(A) ≤

n

i=n−k+1

|λi(A)| for every 1 ≤ k ≤ n

can be deduced easily (exercise!). Equality is achieved for k = n and we have

(1.2)

n

i=1

|λi(A)| = | det(A)| = | det(A)||det(A∗)| = | det(

√

AA∗)| =

n

i=1

si(A).

One may deduce from Weyl’s inequalities that (see [HJ, Theorem 3.3.13])

(1.3)

n

i=1

|λi(A)|2

≤

n

i=1

si(A)2

=

Tr(AA∗)

=

n

i,j=1

|Ai,j|2.

Since s1(·) = ·2→2 we have for any A, B ∈ Mn(C) that

(1.4) s1(AB) ≤ s1(A)s1(B) and s1(A + B) ≤ s1(A) + s1(B).

We define the empirical eigenvalues and singular values measures by

μA :=

1

n

n

k=1

δλk(A) and νA :=

1

n

n

k=1

δsk(A).

Note that μA and νA are supported respectively in C and R+. There is a rigid

determinantal relationship between μA and νA, namely from (1.2) we get

log |λ| dμA(λ) =

1

n

n

i=1

log |λi(A)|

=

1

n

log | det(A)|

=

1

n

n

i=1

log(si(A))

= log(s) dνA(s).

This identity is at the heart of the Hermitization technique used in section 4.

The singular values are quite regular functions of the matrix entries. For in-

stance, the Courant-Fischer formulas imply that the map A → (s1(A),...,sn(A))

is 1-Lipschitz for the operator norm and the

∞

norm: for any A, B ∈ Mn(C),

(1.5) max

1≤i≤n

|si(A) − si(B)| ≤ s1(A − B).

Recall that Mn is a Hilbert space for the dot product A · B = Tr(AB∗). The

associated norm ·HS, called the Hilbert-Schmidt norm3, satisfies to

(1.6) A

2

HS

=

Tr(AA∗)

=

n

i=1

si(A)2

= n

s2

dνA(s).

In the sequel, we say that a sequence of (possibly signed) measures (ηn)n≥1 on

C (respectively on R) tends weakly to a (possibly signed) measure η, and we denote

ηn η,

3Also

known as the trace norm, the Schur norm, or the Frobenius norm.