Below are three subgroups of , the group of invertible
matrices with real entries.
To prove that a subset of a group is a subgroup, we need to show that:
- It contains the identity element.
- It is closed under the group operation.
- It is closed under inverses.
Orthogonal group vs. special orthogonal group
The orthogonal group has the requirement that .
This means that the columns of are orthogonal to each other
and that they all have length 1. This is quite a strong requirement
and you may wonder what is left to distinguish this group from the
special orthogonal group. The answer is that the special orthogonal group
has the additional requirement that the determinant of is
positive 1. What is not in the special orthogonal group
are all the matrices with determinant -1.
is a subgroup of .
Proof.
The identity matrix for has determinant 1, so
it is in . For any matrices and
in , . An so if
, then . So is in
.
As , if has determinant 1, then so too must
. So we have closure under matrix multiplication and inverses,
and the identity is present.
is a subgroup of .
Proof idea. An orthogonal matrix is one whose columns are orthogonal to
each other. The main task is to show that the matrix multiplication of two
orthogonal matrices is also orthogonal. Let and be orthogonal
matrices and consider . Let's be concrete and fix the matrix
dimensions to be . Each column of is a linear
combination of the columns of with coefficients given by the columns
of . Consider testing the dot product of two columns and
of . will be
where is the th column of . Similarly, will
be . When we take the dot product of
and , out of the 9 sub-products that make up the dot product,
the only terms that will survive are those like ,
as the other terms will be orthogonal. But is
just , and so the dot product of
and is just
.
Now remember that which is another way of saying that
the dot product of any column with itself is 1, and the dot product of any
two different columns is 0. So and so the dot product of and is
.
But this is just the dot product of the first column of with the
second column of , which is 0. For the same two columns of ,
the dot product will be 1. This is what we wanted in order for to be
orthogonal.
Proof.
If and are subgroups of a group , then their
intersection is also a subgroup of .
Applying this rule to and
as subgroups of ,
we see that is a subgroup of
also.