Page 142 - Matrices theory and applications

7.5. The Orthogonal Groups O(p, q)
7.5.1 Notable Subgroups of O(p, q)
We assume here that p, q ≥ 1, so that O(p, q) has four connected
components. We first describe them.
Let us recall that if M ∈ O(p, q) reads blockwise

A
M =
D
C B , 125
T
T
where A ∈ M p (IR), etc. Then A A = C C + I p is larger than I p as a
T T
symmetric matrix, so that det A cannot vanish. Similarly, D D = B B+I q
shows that det D does not vanish. The continuous map M → (det A, det D)
2
thus sends O(p, q)to IR × IR (in fact, to (IR \ (−1, 1)) ). Since the sign
∗
∗
∗
map from IR to {−, +} is continuous, we may thus define a continuous
function
σ 2 2
O(p, q) →{−, +} ∼ (ZZ/2ZZ) ,
M → (sgn det A, sgn det D).
The diagonal matrices whose diagonal entries are ±1belongto O(p, q). It
follows that σ is onto. Since σ is continuous, the preimage G α of an element
2
α of {−, +} is the union of some connected components of O(p, q); let n(α)
be the number of these components. Then n(α) ≥ 1(σ being onto), and
n(α) equals 4, the number of connected components of O(p, q). Since

α
there are four terms in this sum, we obtain n(α) = 1 for every α. Finally,
2
the connected components of O(p, q)are the G α ’s, where α ∈{−, +} .
The left multiplication by an element M of O(p, q) is continuous, bijec-
tive, whose inverse (another multiplication) is continuous. It thus induces a
permutation of the set π 0 of connected components of O(p, q). Since σ in-
2
duces a bijection between π 0 and {−, +} , there exists thus a permutation
2
q M of {−, +} such that σ(MM )= q M (σ(M )). Similarly, the multiplica-

tion at right by M is an homeomorphism, allowing to define a permutation
2

p M of{−, +} such that σ(MM )= p M (σ(M)). The equality

p M (σ(M)) = q M (σ(M ))
shows that p M and q M actually depend only on σ(M). In other words,

σ(MM ) depends only on σ(M)and σ(M ). A direct evaluation in the
special case of matrices in O(p, q)∩O n (IR) leads to the following conclusion.
Proposition 7.5.2 (p, q ≥ 1) The connected components of G = O(p, q)
are the sets G α := σ −1 (α), defined by α 1 det A> 0 and α 2 det D> 0,when
2
amatrix M is written blockwise as above. The map σ : O(p, q) →{−, +}
is a surjective group homomorphism; that is, σ(MM )= σ(M)σ(M ).In

particular:
1. G −1 = G α ;
α
2. G α · G α = G αα .