Page 96 - Matrices theory and applications

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(a) Show that · y is a norm on CC for every y =0.
(b) Let N y be the norm induced by · y . Show that N y ≤ · .
(c) We say that · is minimal if there exists no other algebra norm
less than or equal to · . Show that the following assertions are
equivalent:
i. · is an induced norm on M n (CC).
ii. · is a minimal norm on M n (CC). 4.6. Exercises 79
iii. For all y = 0, one has · = N y .
24. (continuation of exercise 23)
Let · be an induced norm on M n (CC).
n
(a) Let y, z = 0 be two vectors in CC . Show that (with the notation
of the previous exercise) · y / · z is constant.
(b) Prove the equality
∗
∗
∗
xy · zt = xt · zy .
∗
25. Let M ∈ M n (CC)and H ∈ HPD n be given. Show that
1 2 2
HMH 2 ≤ H M + MH 2 .
2
2
26. We endow IR with the Euclidean norm · 2 ,and M 2 (IR)with the
induced norm, denoted also by · 2. We denote by Σ the unit sphere of
T
M 2 (IR): M ∈ Σisequivalentto M 2 =1, that is, to ρ(M M)= 1.
Similarly, B denotes the unit ball of M 2 (IR).
Recall that if C is a convex set and if P ∈ C,then P is called an
extremal point if P ∈ [Q, R]and Q, R ∈ C imply Q = R = P.
(a) Show that the set of extremal points of B is equal to O 2 (IR).
(b) Show that M ∈ Σ if and only if there exist two matrices P, Q ∈
O 2 (IR)and anumber a ∈ [0, 1] such that

a 0
M = P Q.
0 1
(c) We denote by R = SO 2 (IR) the set of rotation matrices, and
by S that of matrices of planar symmetry. Recall that O 2 (IR)is
the disjoint union of R and S. Show that Σ is the union of the
segments [r, s]as r runs over R and s runs over S.

(d) Show that two such “open” segments (r, s)and (r ,s )are either
disjoint or equal.
(e) Let M, N ∈ Σ. Show that M − N 2 = 2 (that is, (M, N)is a
diameter of B) if and only if there exists a segment [r, s](r ∈R
and s ∈S) such that M ∈ [r, s]and N ∈ [−r, −s].