That's true. Integers of this form a+b*sqrt(5) are called quadratic integers (https://en.wikipedia.org/wiki/Quadratic_integer https://en.wikipedia.org/wiki/Quad

That's true. Integers of this form a+b*sqrt(5) are called quadratic integers (https://en.wikipedia.org/wiki/Quadratic_integer https://en.wikipedia.org/wiki/Quadratic_integer), and define a ring for which you can define addition and multiplication rules: (a+b*sqrt(5)) + (c+d*sqrt(5)) = (a+c) + (b+d)*sqrt(5) (a+b*sqrt(5)) * (c+d*sqrt(5)) = (ac+5bd) + (ad+bc)*sqrt(5) Note that seeing the operations this way reduce a bit the number of operations compared to the matrix representation you provided: a multiplication using the formula above require 5 integer multiplications instead of 8 with the matrix [[a, 5b], [b, a]]. But this is only a constant factor improvement: the algorithm complexity is still log(n). And the same result can be achieved by using the special structure of the matrix: for your representation, the matrix are all toeplitz matrices, and multiplying those kind of matrices require less operations than full generic matrix multiplication. For the representation in the original article, If I remember correctly, the matrices are triangular, and again you can multiply them more efficiently using this specificity.