Page 151 - Matrices theory and applications

7. Exponential of a Matrix, Polar Decomposition, and Classical Groups
134
iii. Compute the derivative of t → (exp −ty)(exp −tx)exp t(x+
y). Finally, prove the Campbell–Hausdorff formula
1

exp(x + y)=(exp x)(exp y) exp [y, x] .
2
(e) In A = M 3 (IR), construct an example that satisfies the above
hypothesis ([x, y]commutes with x and y), where [x, y]is
nonzero.
17. Show that the map
H → f(H):= (iI n + H)(iI n − H) −1
induces a homeomorphism from H n onto the set of matrices of U n
whose spectrum does not contain −1. Find an equivalent of f(tH) −
exp(−2itH)as t → 0.
18. Let G be a group satisfying the hypotheses of Proposition 7.3.2.
(a) Show that G is a Lie algebra, meaning that it is stable under the
bilinear map (A, B) → [A, B]:= AB − BA.
(b) Show that for t → 0+,
3
2
exp(tA)exp(tB)exp(−tA)exp(−tB)= I n + t [A, B]+ O(t ).
Deduce another proof of the stability of G by [·, ·].
(c) Show that the map M → [A, M] is a derivation, meaning that
the Jacobi identity
[A, [B, C]] = [[A, B],C]+ [B, [A, C]]
holds.
19. In the case p =1, q ≥ 1, show that G ++ ∪ G +− is the set of matrices
M ∈ O(p, q) such that the image under M of the “time” vector
T
(1, 0,... , 0) belongs to the convex cone whose equation is
$
2
2
x 1 > x + ··· + x .
n
2
20. Assume that p, q ≥ 1 and consider the group O(p, q). Define G 0 :=
G ++ .Since −I n ∈ O(p, q), we denote by (µ, β) the indices for which
−I n ∈ G µ,β .
If H ∈ GL n (IR), denote by σ H the conjugation M → H −1 MH.
(a) Let H ∈ G be given. Show that σ H (or rather its restriction to
G 0 ) is an automorphism of G 0 .
(b) Let H ∈ M n (IR) be such that HM = MH for every M ∈ G 0 .
Show that HN = NH for every N ∈G. Deduce that H is a
homothety.
(c) Let H ∈ G. Show that there exists K ∈ G 0 such that σ H = σ K
if and only if H ∈ G 0 ∪ G µ,β .