Clifford algebras and spin groups are important in a variety of applications, and their realizations as matrix algebras have been well studied. Often, however, the spin group Spin+(p, q) is represented as a subgroup of a matrix algebra isomorphic to the even subalgebra of Cl(p, q), rather than the matrix realization of Cl(p, q) itself. This is unsatisfactory for any applications that require the spin group to be explicitly given as a subset of the Clifford algebra. In this thesis we consider pairs p, q ∈ N with 3 ≤ p + q ≤ 6 and give a full construction of Cl( p, q) and Spin+(p, q) within the same matrix algebra, along with the 1-vectors and the Clifford conjugation, reversion, and grade maps on Cl( p, q). Our approach utilizes iterative techniques for constructing Clifford algebras, characterization of when an even dimensional real or complex matrix represents a complex or quaternionic linear transformation, properties of Kronecker products, and the homomorphism between M(4, R) and H ⊗ H, the space of quaternion tensor products, to simplify computations.;A natural application of these constructions is matrix exponentiation, and so we give a full description of how Spin+( p, q) and its Lie algebra Spin+( p, q) can be used to compute the exponential of a matrix in so(p, q,R). For so (n, R), n = 5, 6, we also include explicit formulations of the double covering map phi : Spin(0, n) → SO( n, R), the Lie algebra isomorphism psi : spin(0, n) → so( n, R), and their inversions. Finally, we provide a full characterization of exponentiation on sp(4) and su(4).
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