A new method to obtain Pythagorean triple preserving matrices (Q1407464)

A Pythagorean triple (PT) is a triple \((a,b,c)\) of natural numbers such that \(a^2+b^2=c^2\). The general expression of such triples is \((m^2-n^2,2mn,m^2+n^2)\), \(m,n\in {\mathbb Z}\). Thus a PT is an integer point of the quadric \(x^2+y^2-z^2=X^tSX=0\subset {\mathbb R}^3\) where \(X^t=(x,y,z)\) and \(S= \text{diag}(1,1,-1)\). A \(3\times 3\)-matrix \(A\) is said to be a PT preserving matrix (PTPM) if \(A^tSA=\rho S\) for some \(\rho \in {\mathbb R}\). The author proposes a new method for obtaining PTPMs covering also the singular case \(\rho =0\). Possible connections with physics and quaternions are suggested. The method is generalized to weighted PTs, i.e. triples \((x,y,z)\) satisfying the condition \(p^2x^2+q^2y^2=p^2q^2z^2\).