4
Orthogonal matrices. Orthogonal groups.
Consider a square matrix
a
. The transposed matrix corresponds to the exchange of the symmetric terms with respect to the diagonal, as shown in the figure:
(38)
We denote the inverse matrix
a -1
It satisfies the relation:
a x a -1 = 1
From now on, we will no longer write the x sign and simply write:
a a -1 = 1
When two bold letters are next to each other, we consider that they automatically correspond to the product of two matrices.
An orthogonal matrix is a matrix whose inverse coincides with its transpose.
(38b)
It can be shown that:
(38c)
so that the determinant of an orthogonal matrix is ± 1.
There are orthogonal matrices of any rank (n,n). They form groups
O(n)
O(n) is the set of orthogonal matrices (n,n).
Consider the matrices:
(39)
These are orthogonal matrices whose determinant is:
det (
g
) = +1
It is a subgroup of the orthogonal group
O(2), called the "special orthogonal group" SO(2).
We have an orthogonal group O(3), composed of (3,3) orthogonal matrices, whose determinant = ± 1. It has a subgroup SO(3) composed of orthogonal matrices whose determinant is + 1.
In four dimensions: we have the orthogonal group O(4) and its subgroup: the special orthogonal group SO(4).
In n dimensions: orthogonal group O(n), composed of (n,n) orthogonal matrices, whose determinant is ± 1. It has a subgroup called the special orthogonal group SO(n), restricted to orthogonal matrices whose determinant is + 1.
It can be shown that the dimension of an orthogonal group is (40)
Application to two-dimensional space: the dimension of the group is 1.
Application to three-dimensional space, the dimension of the group is three (the three Euler angles).
Application to four-dimensional space, the dimension becomes six.
We have introduced the oriented special Euclidean group SE(2):
(41)
which combines rotations and translations.
Let us denote:
(42)
Then we can write the matrix and its action on space:
(43)
Note:
(44)
In our two-dimensional flat space, in our plane, we find objects such as:
(45)
Considering these particular objects:
(46)
they belong to the same species. If I take any pair of these objects, I can find an element of the group that maps the first onto the second, and vice versa.
The second subset of objects:
(47)
belongs to another species.
The third one too:
(48)
But:
(49)
I cannot find any combination of rotation a plus translation
c
that maps one onto the other.
Can we modify the oriented Euclidean group in order to make this possible?
Index Dynamic Groups Theory