| 2 |
3 - Third axiom of group theory:
Any element of the group must possess its inverse, written g-1, defined by:
(15)
g x g-1 = g-1 x g = 1
In our example:
(16)
i.e.: b = - a or:
(17)
g-1 ( a ) = g ( - a )
Here the calculation of the inverse matrix is trivial.
What is the condition for a given square matrix to possess its inverse?
...To any square matrix we can associate a scalar called determinant. For the definition see any book devoted to linear calculus. This determinant is denoted: det ( g )
In addition we have a general theorem:
det (g1 x g2) = det (g1 ) x det (g1 )
The determinant of a diagonal matrix is:
(18)
As a consequence:
det ( 1 ) = 1
for 1 is a diagonal matrix.
From the definition of the inverse of a matrix:
g x g-1 = g-1 x g = 1
Then:
(19)
det ( g x g-1 ) = det (g) x det (g-1 ) = 1
...If det (g) = 0 the condition (19) cannot be satisfied. Sets of matrices whose particular elements have null determinant do not satisfy the third axiom, and cannot form a group.
By the way:
(20)
4 - Fourth axiom of group theory:
The multiplication must be associative, i.e.: .
(21)
( g1 x g2 ) x g3 =g1 x ( g2 x g3 )
Matrix multiplication is basically associative.
Dimension of a group:
...As we will see, a group may act on a space whose points are described by column-vectors. For an example space-time points (called "events"):
(22)
...This is a four dimensions space. Different groups may act on it. But the dimension of a group has nothing to do with the dimension of the space it acts on.
The dimension of a group (of matrices) is the number
of parameters which define these square matrix.
We have given an example of matrices, defined by a single parameter
a
So that the dimension of this group is one.
Notice that:
(22-bis)
Remark:
All groups of matrices are not commutative, although the group we studied possesses this property:
(23)
If such group acts on a column vector, corresponding to a 2d space:
(23 bis)
it corresponds to rotation around a fixed point, in a plane:
(23 ter)
This operation is obviously commutative.
You will tend to say: "like all rotations groups".
...You're wrong. Consider the rotations around axis passing by a given point O. Combine two successive rotations, around different axis. This is not commutative. Exercise: show that, using orthogonal axis system (OX, OY, OZ), combined rotations around these axis is not a commutative operation.. Take any object.
- Make a rotation +90° around OX, then a rotation +90° around OZ
Return to initial conditions and:
- Make a rotation +90° around OZ, then a rotation +90° around OX
Compare the results.
Group's action.
...A group G is composed by square matrixes g . They can be multiplied. We will say that a group may act on itself .
The group may also act on a space, made of points, described by column vectors. Example:
(24)
If we write:
(25)
the action of the group on this space becomes:
(26)
g x r
...In this peculiar case the action on space identifies to the simple matrix multiplication. But the concept of an action is much more general.
