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.
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