How to find all fixed points for this problem?

Question

Find all fixed points of the below function $$f:X\rightarrow X$$. $$X=R^N$$ and $$d(x,y)\equiv\Vert x-y\Vert_2$$ and $$f(x)\equiv Ax$$ where $$A\equiv \begin{bmatrix} 1 & 1 \\ 0 &1 \\ \end{bmatrix}$$ Is it right to calculate all the intersection points of y=x and y=Ax for this problem? How to justify this problem?

Answer 1

Fixed points are defined by $Ax=x$. This gives $(x+y,y)=(x,y)$ which is true iff $y=0$. So points on the $x-$ axis are precisely the fixed points of $A$.