Page 152 - Matrices theory and applications

7.8. Exercises
135
21. A topological group is a group G endowed with a topology for which
the maps (g, h) → gh and g → g
topological group, the connected component of the unit element is
a normal subgroup. Show also that the open subgroups are closed.
Give an example of a closed subgroup that is not open.
2n
with CC by the map
22. One identifies IR
n −1 are continuous. Show that in a
x
→ x + iy.
y
˜
Therefore, every matrix M ∈ M 2n (IR) defines an IR-linear map M
n
form CC into itself.
(a) Let

A B
M = ∈ M 2n (IR)
C D
be given. Under what condition on the blocks A, B, C, D is the
˜
map MCC-linear?
˜
(b) Show that M → M is an isomorphism from Sp ∩O 2n onto U n .
n