ABSTRACT
In this chapter, we introduce the reader to several fundamental concepts. We begin by introducing matrices, and operations that pertain to them. Matrix addition and scalar multiplication, for example, allow us to demonstrate that all n×n matrices with entries from the number field F belong to a vector space, denoted V(n × n,F). Since the definition of vector spaces requires fields and is constructed by adding structure to a group, we briefly introduce these concepts. In this chapter, the theory of Lie algebras and Lie groups over the real number field R will be introduced first, as these are easier to handle. However, vector space and group theory applications to quantum mechanics require the extension of these concepts to the complex field C. The definition of matrix product allows us to add algebraic structure to the vector space V (n × n,R) and generate the general linear algebra of order n. The reader is guided to learn the noncommutative nature of the matrix product from which antisymmetrized and symmetrized product definitions can be constructed. The antisymmetric commuted product and the Jacobi identity, are used to introduce two important Lie algebras: so (n,R) and sp (n,R). Using Taylor’s theorem, we construct the corresponding special orthogonal groups, SO (n,R), and the symplectic groups, Sp (n,R). The complex field equivalent of the so (n,R) algebra and the special orthogonal group are su (n,C) and SU (n,C), respectively, the special unitary Lie algebra and group. We revisit the general linear group Gl (n,R) on the real number field and Gl (n,C) on the complex number field, as groups of continuous linear transformations on two sets of vector spaces, V (1 × n,R) and V (n × 1,R). The Lie groups are subgroups of these general groups over the respective fields. Diagonalization is introduced as a way to produce subgroups of SO (n,R) for symmetric real matrices and SU (n,C) for complex Hermitian matrices.
