The groups one studies in geometry are usually groups of matrices. The simplest example is the group of inversible matrices. This is just the set of nonzero real numbers with their standard multiplication. The higher dimensional analogue is the group of inversible matrices provided with their usual multiplication.

Before you go on, make sure you can **effortlessly** solve the following
exercises:

and

*If you cannot solve these problems in a very short period of time, then
you are not ready for this course. Contact your instructor.*

Besides the group of invertible matrices of real numbers, we also have the group of invertible matrices of complex numbers. Notice that is a subgroup of . In fact, almost every interesting group in geometry is a subgroup of . Here are some examples:

*The special linear group.* The set of real (resp. complex)
matrices having determinant one is denoted by
(resp.
) and is called the special linear
group. If
, then the transformation from
to itself defined by
preserves volume and
orientation.

*The orthogonal and the special orthogonal group.* A square matrix is
said to be orthogonal if its transpose is equal to its inverse. The
set of
orthogonal matrices is denoted by . The
subset of consisting of matrices of determinant one is
denoted by . They are respectively the orthogonal and the
special orthogonal group.

- Show that if is any matrix and is its transpose, then . Conclude that if , then .
- Show that if is an orthogonal matrix, the distance between and equals the distance between and .
- Show that the determinant of an orthogonal matrix is either or

*The unitary and the special unitary group.* If is a
square matrices of complex numbers, the *adjoint* of ,
denoted by , is the result of transposing and then
conjugating all of its entries. For example

- Show that if is any
complex matrix and
is its adjoint, then
. Conclude
that if is unitary, then
- Show that if is a unitary matrix, the distance between and equals the distance between and .
- Show that the determinant of an unitary matrix is a complex number of modulus one.

- Show that the set of all real matrices that commute with can be naturally identified with the set of complex numbers.
- By analogy with the previous item, show that the group can be seen as a subgroup of .
- Define the
*symplectic product*of two vectors on by . Show that the symplectic product is antisymmetric and nondegenerate. - Define the
*linear symplectic group*as the set of all real matrices satisfying . Show that the intersection of and equals .

- Show that .
- Show that if has zero trace, then has zero trace.

The following exercise was taken from the book of Abraham and Marsden on the foundations of mechanics (see [1]).

- Show that is a vector space and compute its dimension.
- Show that if and , then .
- The three
*Pauly spin matrices*from quantum mechanics areandShow that the matrices , form a basis of with the commutation relations - Indentify
with by assigning to each vector
the matrix
- Show that the determinant of equals .
- Let be in and consider the transformation from to itself defined by . Show that is a linear transformation that preserves distances and has determinant one.
- Show that if and are in , then if and only if .

From this exercise we conclude that the group , which we saw earlier identified with the -dimensional sphere, has a natural -to- correspondence with the group of rotations . To this quirk of nature we owe the spin of electrons and a pretty little trick due to Dirac and which we shall describe in chapter 5.