Compound Matrices and Pfaffians: A Representation of Geometric Algebra

Abstract

We consider the Clifford algebra Cl _n (F) where the field F is the real R or the complex numbers C. It is well known that an m-form x ₁/\⋯/\ x _m can be represented by the mth compound matrix of the n-by-m matrix X := [x ₁, ⋯, x _m ] ∈ F^n×m relative to the basis {θ ₁, θ ₂, ⋯, θ _2n } 2254; 1, e ₁, ⋯, e ₁Λ⋯Λe _n } of the underlying Grassmann algebra G _n (F). Since the Clifford product ● is related to the Grassmann product A via x ● y = x Λ y + x ^T y, x, y ∈ Fⁿ, the question of a corresponding representation of the Clifford product x ₁ ·⋯· x _m arises in a natural way. We will show that the Clifford product of an odd (even) number of vectors corresponds to a linear combination of forms of odd (even) grade where the coefficients of these linear combinations are Pfaffians of certain matrices which can be understood as the skew symmetric counterpart of the corresponding Gramians. Based on this representation we calculate the mth Clifford power $\underline x ^m :{\rm{ = }}\overbrace {x \bullet \cdots \bullet x}^m$ of a vector x ∈ F _n which enables the extension of an analytical function f : F → F to their corresponding Clifford function f:F ⁿ → Cl _n(F).