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## Important examples from linear algebra

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:

Exercise 1.4   Multiply the matrices

and

Exercise 1.5   Mentally compute the inverse of the matrix

Exercise 1.6   Compute the determinant of

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.

Exercise 1.7 (05)   Let us denote the scalar or inner product of two vectors and in by .
1. Show that if is any matrix and is its transpose, then . Conclude that if , then .
2. Show that if is an orthogonal matrix, the distance between and equals the distance between and .
3. 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

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

Exercise 1.8 (05)   Let us define the Hermitian product of two vectors and in by .
1. Show that if is any complex matrix and is its adjoint, then . Conclude that if is unitary, then

2. Show that if is a unitary matrix, the distance between and equals the distance between and .
3. Show that the determinant of an unitary matrix is a complex number of modulus one.

Exercise 1.9 (*10)   In what follows will denote the identity matrix and is the matrix .
1. Show that the set of all real matrices that commute with can be naturally identified with the set of complex numbers.
2. By analogy with the previous item, show that the group can be seen as a subgroup of .
3. Define the symplectic product of two vectors on by . Show that the symplectic product is antisymmetric and nondegenerate.
4. Define the linear symplectic group as the set of all real matrices satisfying . Show that the intersection of and equals .

Exercise 1.10 (15)   Show that the group can be identified with the -dimensional unit sphere in .

Definition 1.3   The trace of a square matrix is the sum of all the elements in its diagonal. The commutator of two matrices and , denoted by , is defined as

Exercise 1.11 (05)   In what follows , , and are matrices with invertible
1. Show that .
2. 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]).

Exercise 1.12 (15)   Let be the set of complex matrices of zero trace such that .
1. Show that is a vector space and compute its dimension.
2. Show that if and , then .
3. The three Pauly spin matrices from quantum mechanics are

and

Show that the matrices , form a basis of with the commutation relations

where equals if is an even permutation of and otherwise.

4. Indentify with by assigning to each vector the matrix

If denotes the vector product of two vectors in , show that

5. Show that the determinant of equals .
6. 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.
7. 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.

Next: Actions of groups on Up: Groups Previous: Motivation and basic definitions
Juan Carlos Alvarez 2000-10-27