Pythagorean Vectors and Rational Orthonormal Matrices
Abstract
A Pythagorean vector is an integer vector having an integer 2-norm. Such vectors are closely related to Pythagorean n-tuples, since n-tuples are the building blocks for Pythagorean vectors. Pythagorean vectors are, in their turn, the building blocks for rational orthonormal matrices. The work in this thesis has a pedagogical application to the QR decomposition of matrices, widely used in Linear Algebra. A barrier for students learning the details of the QR decomposition of a given matrix A is the occurrence of square-roots that cannot be simplified during the application of the two standard algorithms, namely the Gram--Schmidt method and Householder transformations. This thesis studies Pythagorean vectors and their application to the construction of exercises and test questions in which a given matrix A can be factored into matrices Q and R, with all arithmetic operations resulting in rational quantities, free from square roots. This freedom from square roots applies to every step of the calculations, and not just the final result.
As a preliminary to QR decomposition, the thesis explores the properties of Pythagorean vectors, including their generation for an arbitrary specified dimension. Pythagorean triples, which correspond to Pythagorean vectors of dimension 2, have been widely and enthusiastically studied in the literature, but higher dimensions have been less studied, and this thesis adds some new observations to previous studies.