The Inverse of a Multivector: Beyond the Threshold $$\varvec{}$$

The algorithm of finding an inverse multivector (MV) numerically and symbolically is of paramount importance in the applied Clifford geometric algebra (GA) \( Cl _{p,q}\). The first general MV inversion algorithm was based on matrix representation of MV. The complexity of calculations and the size of the answer in a symbolic form grow exponentially with the dimension \(n=p+q\). The breakthrough occurred when Lundholm and then Dadbeh found compact inverse formulas up to dimension \(n\le 5\). The formulas were constructed in a form of Clifford product of the initial MV and carefully chosen grade-negation counterparts. In this report we show that the grade-negated self-product method can be extended beyond the threshold \(n=5\) if, in addition, properly constructed linear combinations of such MV products are used. In particular, we present compact explicit MV inverse formulas for algebras of vector space of dimension \(n=6\) and show that they embrace all lower dimensional cases as well. For readers convenience, we have also given various MV formulas in a form of grade negations when \(n\le 5\).