Applying Matrix Computations to Geometric Algebra

The primary purpose of this paper is to explore the connection between matrix computations and the mathematical objects of geometric algebra. We refer to the specific type of objects that were examined as `multiblades', which are a generalization of a special case of the multivectors of geometric algebra. Decompositions using unitary matrices are key tools in matrix computations, and this motivated an exploration of the possibilities for unitary decompositions in geometric algebra. Some grades of multiblades have matching matrix representations. As a result, methods for singular value decomposition, and ideas such as the Cayley-Hamilton Theorem, can be applied to geometric algebra.

These results are not new, but were motivated by a separate study into some matrix operators and a desire to treat multivectors where the subcomponent fields are matrices rather than scalars or complex numbers.