Page 73 - Matrices theory and applications

3. Matrices with Real or Complex Entries
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3. Show that a triangular and normal matrix is diagonal. Deduce that
∗
if U TU is a unitary trigonalization of M,and if M is normal, then
T is diagonal.
4. For A ∈ M n (IR), symmetric positive definite, show that
max |a ij | =max a ii .
i,j≤n i≤n
5. Given an invertible matrix

a b
M = ∈ GL 2 (IR),
c d
2
define a map h M from S := CC ∪{∞} into itself by
az + b
h M (z):= .
cz + d
(a) Show that h M is a bijection.
(b) Show that h : M → h M is a group homomorphism. Compute its
kernel.
(c) Let H be the upper half-plane, consisting on those z ∈ CC with
z> 0. Compute h M (z)in terms of z and deduce that the
subgroup
+
GL (IR):= {M ∈ GL 2 (IR) | det M> 0}
2
acts on H.
(d) Conclude that the group PSL 2 (IR):= SL 2 (IR)/{±I 2 }, called
the modular group,actson H.
(e) Let M ∈ SL 2 (IR) be given. Determine, in terms of Tr M,the
number of fixed points of h M on H.

N
6. Show that the supremum of a family of convex functions on IR is
convex. Deduce that the map M → λ n (largest eigenvalue of M)
defined on H n is convex.

7. Show that M ∈ M n (CC) is normal if and only if there exists a unitary
∗
matrix U such that M = MU.
8. Show that in M n(CC) the set of diagonalizable matrices is dense. Hint:
Use Theorem 3.1.3.

9. Let (a 1 ,... ,a n )and (b 1 ,... ,b n ) be two sequences of real numbers.
Find the supremum and the infimum of Tr(AB)as A (respectively B)
runs over the Hermitian matrices with spectrum equal to (a 1 ,... ,a n )
(respectively (b 1 ,... ,b n )).

10. (Kantorovich inequality)