Polar and Givens Decomposition and Inversion of the Indefinite Double Covering Map

Algorithmic methods for the explicit inversion of the indefinite double covering maps are proposed. These based on either the Givens decomposition or the polar decomposition of the given matrix in the proper, indefinite orthogonal group SO+(p, q). As a by product we establish that the pre-image in covering group, of a positive matrix in SO+(p, q), can always be chosen to be itself positive definite. These methods solve the given system by either inspection or by inverting the associated Lie algebra isomorphism and computing certain exponential explicitly. The techniques are illustrated for (p, q) ∈ {(2, 1),(4, 1),(5, 1)}. We also develop explicit matrix form for the covering map for the cases(p,q) in (2,3) and (2,4) . In this work we provide explicit algorithms to invert these double cover maps for specific values of (p, q) (though the general techniques extend to all p, q). More precisely, given the double covering map, Φp,q : Spin+ (p, q) → SO+(p, q) and an X ∈ SO+(p, q) (henceforth called the target), we provide algorithms to compute the matrices ±Y , in the matrix algebra that the even sub algebra of Cl(p, q) is isomorphic to, satisfying Φp,q(±Y ) = X