Incommensurability 
Dynamic V
Let \(B\) be the inverse of the matrix \(A\). Note that the eigenspaces of \(A\) are also eigenspaces of \(B\), but the eigenvalues of the matrix \(B\) are the inverses of the eigenvalues of \(A\). Therefore, if the initial point lies on neither of the straight lines \(x=\sqrt{2}y\) and \(x=-\sqrt{2}y\), then all the iterates will also be outside those lines, getting farther from the line \(x=-\sqrt{2}y\) and closer to the line \(x=\sqrt{2}y\), that is, approaching the values of the diagonal and the side of a square, in case both are positive. If the coordinates of the initial point are positive integers, this allows us to obtain approximations to the ratio between the diagonal and the side of a square using rational numbers.
For example, take \((x_{0},y_{0})=(1,1)\). We have: \[\begin{array}{ll} (x_{1},y_{1})=(3,2) & \;\;\;\frac{3}{2}=1,5\\ (x_{2},y_{2})=(7,5) & \;\;\;\frac{7}{5}=1,4\\ (x_{3},y_{3})=(17,12) & \;\;\;\frac{17}{12}=1,41666...\\ (x_{4},y_{4})=(41,29) & \;\;\;\frac{41}{29}=1,41379...\\ (x_{5},y_{5})=(99,70) & \;\;\;\frac{99}{70}=1,41428... \end{array}\] where \(\sqrt{2}=1,41421356...\)
Indeed, taking the sequence \(r_{n}=\frac{x_{n}}{y_{n}}\), we have \(\lim r_{n}=\sqrt{2}\), where \(r_{n}\in\mathbb{Q},\; \forall n\in\mathbb{N}.\)
Why does this happen?
