Generalized Pythagorean Triples and Pythagorean Triple Preserving Matrices

Abstract

Traditionally, Pythagorean triples (PT) consist of three positive integers, $(x, y, z) \in \mathbb{Z}^3_+$, such that $x^2 + y^2 = z^2$, and Pythagorean triple preserving matrices (PTPM) $A$ are $3 \times 3$ matrices with entries in the real numbers $\R$, such that the product $(x, y, z)A$ is also a Pythagorean triple. In this paper, we study PT and PTPM from the view of projective geometry, and extend the results concerning PT and PTPM from integers to any commutative ring with identity. In particular, we use the method of polynomial parametrization for projective conics to obtain the general form of PT over any commutative ring with identity. In addition, we view the PTPM as projective transformations and formulate the general form of a PTPM over any commutative ring with identity.