Page 148 - Matrices theory and applications

(c) Let us define Q(t):= [P (t),P(t)]. Show that P (t)=
[Q(t),P(t)].
(d) Let t 0 ∈ I be given. Show that the differential equation U = QU
possesses a unique solution in I such that U(t 0 )= I n . Show that
P(t)= U(t)P(t 0 )U(t)
5. Show that the set of projectors of given rank p is a connected subset
in M n (CC). −1 . 7.8. Exercises 131
6. (a) Let A ∈ HPD n and B ∈ H n be given. Show that AB is di-
agonalizable with real eigenvalues (though it is not necessarily
Hermitian). Show also that the sum of the multiplicities of the
positive eigenvalues (respectively zero, respectively negative) is
the same for AB as for B.
(b) Let A, B, C be three Hermitian matrices such that ABC ∈ H n .
Show that if three of the matrices A, B, C, ABC are positive
definite, then the fourth is positive definite too.
7. Let M ∈ GL n (CC)be given and M = HQ be its polar decomposition.
Show that M is normal if and only if HQ = QH.
8. The deformation of an elastic body is represented at each point by a
+
square matrix F ∈ GL (IR) (the sign + expresses that det F> 0).
3
+
More generally, F ∈ GL (IR) in other space dimensions. The density
n
+
of elastic energy is given by a function F → W(F) ∈ IR .
(a) The principle of frame indifference says that W(QF)= W(F)
+
for every F ∈ GL (IR) and every rotation Q. Show that there
n
+
exists a map w : SPD n → IR such that W(F)= w(H), where
F = QH is the polar decomposition.
(b) When the body is isotropic, we also have W(FQ)= W(F), for
+
every F ∈ GL (IR) and every rotation Q. Show that there exists
n
n
amap φ : IR → IR + such that W(F)= φ(h 1 ,... ,h n ), where
the h j are the entries of the characteristic polynomial of H.In
other words, W(F) depends only on the singular values of F.
9. We use Schur’s norm A =(Tr A A) 1/2 .
∗
(a) If A ∈ M n (CC), show that there exists Q ∈ U n such that A −
Q ≤ A − U for every U ∈ U n . We shall define S := Q −1 A.
We therefore have S − I n ≤ S − U for every U ∈ U n .
(b) Let H ∈ H n be a Hermitian matrix. Show that exp(itH) ∈ U n
for every t ∈ IR. Compute the derivative at t =0 of
2
t → S − exp(itH)
and deduce that S ∈ H n .
(c) Let D be a diagonal matrix, unitarily similar to S.Show that
D−I n ≤ DU −I n for every U ∈ U n . By selecting a suitable
U, deduce that S ≥ 0 n .