Given a rule to obtain a new $x,y$ positions from an initial $x_0,y_0$

The rule is

$ binom{x’}{y’} = begin{pmatrix}

v_x & -v_y \

v_y & v_x

end{pmatrix}

binom{x}{y}$

$x’ = v_x x – v_y y$

$y’ = v_y x + v_x y$

$f(x,y) = (v_x x – v_y y, v_y x + v_x y)$

Now let’s say I wanted to use this rule to update a position $x_0,y_0$

How can I use repeated doubling to calculate the position after updating it $n$ times without having to manually update the position $n$ times

For instance lets say we started at position $(-10, 0) $ and we wanted to update this position 5 times and our given value for $(u_x,u_y) = (frac{1}{2},frac{1}{2})$

after the first update(n=1) the position would be $(-5,-5)$ and then the 5th update(n=5) the position would be $( frac{5}{4}, frac{5}{4}) $

How could I calculate the position for $( frac{5}{4}, frac{5}{4}) $ without having to calculate n = 2 or n= 3 or n= 4 or simply just using the concept of repeated doubling