Quaternion matrix inverses and linear equation solutions are often computed by transforming a given A ∈ Q~(NxN) quaternion matrix into an equivalent A ∈ R~(4Nx4N) 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 R~(4Nx4N) matrix representation that requires only four R~(NxN) matrix inversions to complete the solution. A two-stage algorithm is presented. First the R~(4Nx4N) matrix is partitioned into four blocks using two R~(2Nx2N) matrices. A partitioned matrix inverse is pre-sented for inverting each R~(2Nx2N) matrix partition. A second level of the partitioning exploits the skew-symmetric sub-structure of the R~(2Nx2N) partitioned matrix inverse solutions. The proposed algorithm improves the computational performance ～2X and minimizes the memory requirements by only requiring processing for four R~(NxN) matrix partitions, when compared to standard algorithms for inverting the R~(4Nx4N) matrix. Example applications are presented for a purely quaternion matrix inversion algorithm, as well as solution algorithms for the R~(2Nx2N) and R~(4Nx4N) variable versions of the algorithms.
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