Algorithms for the Polar Decomposition in Certain Groups and the Quaternions

TL;DR

This paper presents simple, constructive algorithms requiring minimal matrix operations for the polar decomposition of matrices in certain 4x4 groups, including Lorentz and symplectic groups, and explores their quaternionic representations.

Contribution

It introduces novel, efficient algorithms for polar decomposition in specific matrix groups and establishes quaternionic representations for these groups, extending prior work on the special orthogonal group.

Findings

Algorithms require no more than 2x2 matrix manipulations

Constructive proof that positive definite matrices are in the connected component of the identity

Quaternionic representations for groups including Lorentz and symplectic groups

Abstract

Constructive algorithms, requiring no more than $2\times 2$ matrix manipulations, are provided for finding the entries of the positive definite factor in the polar decomposition of matrices in sixteen groups preserving a bilinear form in dimension four, including the Lorentz and symplectic groups. This is used to find quaternionic representations for these groups analogous to that for the special orthogonal group.This is achieved by first characterizing positive definite matrices in these groups. For the groups whose signature matrix is a symmetric matrix in the quaternion tensor basis for 4x4 matrices, a completion procedure based on this observation leads to said computation of the polar decomposition, while for the Lorentz group this is achieved by passage to its double cover. Amongst byproducts we mention an elementary and constructive proof showing that positive definite matrices in such groups belong to the connected component of the identity.