Skew-Symmetric Algorithm for Inverting NxN Quaternion Equations Using NxN Real Variable Matrix Arithmetic

Abstract

Quaternion matrix inverses and linear equation solutions are often computed by transforming a given A ∈ QN xN quaternion matrix into an equivalent \(A \in {\cal R}^{4N x4N}\) real matrix. The transformation process is well-known, but as N becomes large the computational costs increase rapidly. This paper presents a multi-level skew-symmetric partitioning algorithm for the \({\cal R}^{4N x4N}\) matrix representation that requires only four \({\cal R}^{4N x4N}\) matrix inversions to complete the solution. A two-stage algorithm is presented. First the \({\cal R}^{4N x4N}\) matrix is partitioned into four blocks using two \({\cal R}^{2N x24N}\) matrices. A partitioned matrix inverse is presented for inverting each \({\cal R}^{2N x2N}\) matrix partition. A second level of the partitioning exploits the skew-symmetric sub-structure of the \({\cal R}^{2N x2N}\) partitioned matrix inverse solutions. The proposed algorithm improves the computational performance ∼ 2X and minimizes the memory requirements by only requiring processing for four \({\cal R}^{N xN}\) matrix partitions, when compared to standard algorithms for inverting the \({\cal R}^{4N x4N}\) matrix. Example applications are presented for a purely quaternion matrix inversion algorithm, as well as solution algorithms for the \({\cal R}^{2N x2N}\) and \({\cal R}^{4N x4N}\) real variable versions of the algorithms.