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Orthogonal matrices. Orthogonal groups.
Take a square matrix a. The transposed matrix corresponds to the exchange of the terms symmetric with respect to the diagonal, as shown in the figure:
(38)
We denote the inverse matrix a-1
It obeys the relation:
a × a-1 = 1
From now on, we will no longer write the × 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)
from which the determinant of an orthogonal matrix is ± 1.
They 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)
They 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 orthogonal matrices (3,3), 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).
n dimensions: orthogonal group O(n), composed of orthogonal matrices (n,n), whose determinant is ± 1. It has a subgroup called the special orthogonal group SO(n), limited 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 plus translation c that allows me to go from one to the other.
Can we modify the oriented Euclidean group in order to make this possible?