I am asked to find the 2x2 matrix A that represents an orthogonal projection onto the line $f(x)=-\sqrt{3}x$, and then find $A^{1001}$. I have found A and tried multiplying it out a couple times to deduce a pattern but haven't got anywhere.
Thinking geometrically, once the projection is performed the first time, wouldn't any subsequent projection be represented by a zero matrix because it's not going anywhere? With that in mind, what approach am I missing to find $A^{1001}$?
Notice that the unit vector $e=\frac12(1,-\sqrt3)$ spans the given line. Now, let $(e_1,e_2)$ the standard basis of $\Bbb R^2$ then the first column of $A$ is given by
$$(e_1\cdot e)e=\frac12 e$$ and the second column of $A$ is given by $$(e_2\cdot e)e=-\frac{\sqrt3}2 e$$ Hence, we have $$A=\begin{pmatrix}\frac14&-\frac{\sqrt3}4\\ -\frac{\sqrt3}4& \frac34\end{pmatrix}$$ Finally, since $A$ is a projection matrix then $A^2=A$ (you can verify it) and so we get $A^{1001}=A$.