Page 150 - Matrices theory and applications

(d) Show that J 4 is the exponential of a matrix of M 4 (IR), but that
J 3 is not.
13. Let A n (CC) be the set of skew-Hermitian matrices of size n.Show
that exp : A n (CC) → U n is onto. Hint:If U is unitary, diagonalize it.
14. (a) Let θ ∈ IR be given. Compute exp B,where
7.8. Exercises 133
0 θ
B = .
−θ 0
(b) Let A n (IR) be the set of real skew-symmetric matrices of size n.
Show that exp : A n (IR) → SO n is onto. Hint: Use the reduction
of direct orthogonal matrices.
15. Let φ : M n (IR) → IR be a nonnull map satisfying φ(AB)= φ(A)φ(B)
1/n
for every A, B ∈ M n (IR). If α ∈ IR,we set δ(α)= |φ(αI n )| .We
have seen, in Exercise 16 of Chapter 3, that |φ(M)| = δ(det M)for
every M ∈ M n (IR).
2
(a) Show that on the range of M → M and on that of M → exp M,
φ ≡ δ ◦ det.
(b) Deduce that φ ≡ δ◦det on SO n (use Exercise 14) and on SPD n .
(c) Show that either φ ≡ δ ◦ det or φ ≡ (sgn(det))δ ◦ det.

16. Let A be a K-Banach algebra (K = IR or CC) with a unit denoted by
0
e.If x ∈ A, define x := e.
(a) Given x ∈ A, show that the series
1
m
x
m!
m∈IN
converges normally, hence converges in A. Its sum is denoted by
exp x.
(b) If x, y ∈ A,[x, y]= xy − yx is called the “commutator” of x and
y.Showthat [x, y] = 0 implies

exp(x + y)=(exp x)(exp y), [x, exp y]= 0.
(c) Show that the map t → exp tx is differentiable on IR,with

d
exp tx = x exp tx =(exp tx)x.
dt
(d) Let x, y ∈ A be given. Assume that [x, y]commuteswith x and
y.
i. Show that (exp −tx)xy(exp tx)= xy + t[y, x]x.
ii. Deduce that [exp −tx, y]= t[y, x]exp −tx.