On congruent circulant matrices

In this work, we investigate congruent circulant matrices and provide a parametrization of congruent matrices in M2(Q). We introduce a matrix analogue of Fermat’s algorithm, which enables the construction of sequences of matrix Pythagorean triples associated with a given 2 × 2 congruent circulant matrix N. Furthermore, we establish a matrix version of Pythagoras’ theorem within the Euclidean vector space Mm(Q); m ≥ 2. Mathematics Subject Classification (MSC2020). 11C20, 15A15, 15A18, 15A20, 15A27, 15A60, 11D72. Keywords: Congruent numbers; Pythagorean triples; Congruent circulant matrix; Matrix Pythagorean triples; Eigenvalues.