Page 149 - Matrices theory and applications

7. Exponential of a Matrix, Polar Decomposition, and Classical Groups
132
(d) If A ∈ GL n (CC), show that QS is the polar decomposition of A.
(e) Deduce that if H ∈ HPD n and if U ∈ U n , U = I n ,then
H − I n < H − U .
(f) Finally, show that if H ∈ H n , H ≥ 0 n and U ∈ U n ,then
H − I n ≤ H − U .
10. Let A ∈ M n (CC)and h ∈ CC. Show that I n − hA is invertible as soon
as |h| < 1/ρ(A). One then denotes its inverse by R(h; A).
(a) Let r ∈ (0, 1/ρ(A)). Show that there exists a c 0 > 0 such that
for every h ∈ CC with |h|≤ r,we have
2
R(h; A) − e hA ≤ c 0 |h| .
(b) Verify the formula
C m − B m =(C − B)C m−1 + ··· + B l−1 (C − B)C m−l + ···
+ ··· + B m−1 (C − B),
and deduce the bound
m
2 c 2 m|h|
R(h; A) − e mhA ≤ c 0 m|h| e ,
when |h|≤ r and m ∈ IN.
(c) Show that for every t ∈ CC,
tA
lim R(t/m; A) m = e .
m→+∞
∗
11. (a) Let J(a; r)be a Jordan block of size r,with a ∈ CC .Let b ∈ CC be
b
such that a = e . Show that there exists a nilpotent N ∈ M r (CC)
such that J(a; r)=exp(bI r + N).
(b) Show that exp : M n(CC) → GL n (CC) is onto, but that it is not
2
one-to-one. Deduce that X → X is onto GL n (CC). Verify that
it is not onto M n (CC).
12. (a) Show that the matrix

−1 1
J 2 =
0 −1
is not the square of any matrix of M 2 (IR).
(b) Show, however, that the matrix J 4 := diag(J 2 ,J 2 ) is the square
of a matrix of M 4 (IR).
Show also that the matrix

J 2 I 2
J 3 =
0 2 J 2
is not the square of a matrix of M 4 (IR).
(c) Show that J 2 is not the exponential of any matrix of M 2 (IR).
Compare with the previous exercise.