2018/07/16 by Francis Adjei, Adjei, Francis, Marcus Cisneros +7
Computer Science · Mathematics · #Algebraic and Geometric Analysis #FOS: Physical sciences #Mathematical Physics (math-ph) #Mathematics and Applications #Matrix Theory and Algorithms
paper · pdf · doi:10.48550/arxiv.1807.06062
openalex publication_date 2018/07/16 · openalex created_date 2022/08/04 · openalex updated_date 2026/07/28
Constructive algorithms, requiring no more than 2\× 2 matrix\nmanipulations, are provided for finding the entries of the positive definite\nfactor in the polar decomposition of matrices in sixteen groups preserving a\nbilinear form in dimension four, including the Lorentz and symplectic groups.\nThis is used to find quaternionic representations for these groups analogous to\nthat for the special orthogonal group.This is achieved by first characterizing\npositive definite matrices in these groups. For the groups whose signature\nmatrix is a symmetric matrix in the quaternion tensor basis for 4x4 matrices, a\ncompletion procedure based on this observation leads to said computation of the\npolar decomposition, while for the Lorentz group this is achieved by passage to\nits double cover. Amongst byproducts we mention an elementary and constructive\nproof showing that positive definite matrices in such groups belong to the\nconnected component of the identity.\n