Algorithms for the Polar Decomposition in Certain Groups and the Quaternion Tensor Square

By : Francis Adjei , Marcus Cisneros , Deep Desai , Samreen Khan , Viswanath Ramakrishna , Brandon Whiteley

Page No: 1-34

Abstract: 
Constructive algorithms, not even requiring 2×2 eigencalculations, are provided for finding the entries of the positive definite factor in the polar decomposition of matrices in many groups. These groups include the indefinite orthogonal groups of signature (1, n – 1) nd (n – 1, 1) and fifteen groups preserving certain bilinear forms in dimension four. The Lorentz group belongs to both classes. Some of these algorithms extend to the indefinite orthogonal groups of arbitrary signature with nominal additional work. These procedures are then used to find quaternionic representations for the four dimensional groups mentioned above, analogous to the representation of the special orthogonal group via a pair of unit quaternions. A key ingredient is a characterization of positive definite matrices in these groups. Two algorithms are proposed for the Lorentz group. The former also works for the group whose signature is (n – 1, 1) and (1, n – 1). The second enables (and is aided by) the inversion of the double covering of the Lorentz group by SL(2, C). A key observation is that the inversion of the covering map, when the target is a positive definite matrix, can be achieved essentially by inspection as we demonstrate. For the group whose signature matrix is I2,2 a completion procedure based on the aforementioned characterization of positivity leads to yet another algorithm for the computation of the polar decomposition. For the other four dimensional groups, explicit isomorphisms provided by quaternion algebra lead to methods for the polar decomposition. As byproducts we give a simple proof of the fact that positive definite matrices in each of these groups belong to the connected component of the identity, find an explicit expression for their logarithm, and provide a characterization of the symmetric matrices in the connected component of the identity of two of these groups in terms of their preimages in the corresponding covering group.

Authors :
Francis Adjei
Department of Mathematics and Statistics,Virginia Polytechnic and State University,Blacksburg, VA.

Marcus Cisneros
Department of Mathematical Sciences, University of Texas at Dallas, Richardson, TX.

Deep Desai
Department of Mathematical Sciences, University of Texas at Dallas, Richardson, TX.

Samreen Khan
Department of Mathematical Sciences, University of Texas at Dallas, Richardson, TX.

Viswanath Ramakrishna
Department of Mathematical Sciences, University of Texas at Dallas, Richardson, TX.

Brandon Whiteley
Department of Mathematical Sciences, University of Texas at Dallas, Richardson, TX.
 

DOI: https://doi.org/10.32381/JCISS.2019.44.1-4.1