Constructive Aspects for the Generalized Orthogonal Group

Abstract

Our main result is a constructive proof of the Cartan-Dieudonn\'{e}-Scherk Theorem in the real or complex fields. The Cartan-Dieudonn\'{e}-Scherk Theorem states that for fields of characteristic other than two, every orthogonality can be written as the product of a certain minimal number of reflections across hyperplanes. The earliest proofs were not constructive, and more recent constructive proofs either do not achieve minimal results or are restricted to special cases. For the real or complex fields, this paper presents a constructive proof for decomposing a generalized orthogonal matrix into the product of the minimal number of generalized Householder matrices.

A pseudo code and the MATLAB code of our algorithm are provided. The algorithm factors a given generalized orthogonal matrix into the product of the minimal number of generalized Householder matrices specified in the CDS Theorem.

We also look at some applications of generalized orthogonal matrices. Generalized Householder matrices can be used to study the form of Pythagorean $n$-tuples and generate them. All matrices can not be factored in a QR-like form when a generalized orthogonal matrix in used in place of a standard orthogonal matrix. We find conditions on a matrix under which an indefinite QR factorization is possible, and see how close we can bring a general matrix to an indefinite QR factorization using generalized Householder eliminations.