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a4104

Orthogonal matrixes. Orthogonal groups.

Take a square matrix a. The transposed matrix corresponds to the commutation of the terms which are symmetric with respect to the diagonal, as shown on the figure :
(38)

We write the inverse matrix a-1
It obeys :

a x a-1 = 1

Right now, we will no longer write the sign x and just write : a a-1 = 1 When two bold letters are side to side, we consider that it corresponds automatically to the multiplication of two matrixes.

An orthogonal matrix is a matrix whose inverse identifies to its transposed.

(38b)

One can show that :
(38c)

so that the determinant of an *orthogonal matrix *is ± 1 .
They are orthogonal matrixes of any rank ( n,n) . They form groups

O(n) O(n) is the set of orthogonal matrixes (n,n).

Consider matrixes :
(39)

They are orthogonal matrixes, whose determinant is :

det ( g) = +1

It is a sub-group of the orthogonal group O(2), which is called "special orthogonal group" SO(2).
We have an orthogonal group O(3), composed by (3,3) orthogonal matrixes, whose determinant = ± 1 . It owns a sub-group SO(3) composed by orthogonal matrixes whose determinant is + 1 .

In four dimensions : we have the orthogonal group O(4) and its sub-group : the special orthogonal group SO(4).

n dimensions : orthogonal group O(n), composed by (n,n) orthogonal matrixes, whose determinant is ± 1 . It owns a sub group, called Special orthogonal SO(n) limited to orthogonal matrixes whose determinant is + 1.

One can show that the dimension of an orthogonal group is (40)

Applying to two dimensional space : the dimension of the group is 1.
Applying to three dimensional space, the dimension of the group is three ( the three Euler's angles ).
Applying to four dimensional space, the dimension becomes six.
We have introduced the oriented Special Eulclid's group SE(2):
(41)

Which combined rotations and translation.
Call :
(42)

Then we can write the matrix and the action on space :
(43)

Remark :
(44)

In our 2d flat space, in our plane, we find objects like :
(45)

Considering these peculiar objects :
(46)

they belong to a species. If I take any couple of those object, I can find and element of the group which carries the first onto the second, and vice-versa.
The second sub-set of object :
(47)

belongs to another species.
The third, too :
(48)

But :
(49)

I cannot fing any combination rotation a plus translation c that put one on the other.
Can we modify oriented Euclid's group in order to make this possible ?

Index Dynamic Groups Theory