Equation of a line about which we are reflecting

Question

Let $A$ be the matrix of a reflection about a line of the euclidean plane (w.r.t. the standard basis). How can I find the equation of the line?

Answer 1

Since an vector $v$ pointing along the line will be unaffected by multiplication by $A$, we have $$ Av = v \\ Av - v = 0 \\ (A-I) v = 0 $$ So you can solve for the kernel of $A-I$ to get a vector $v = (p, q)$ and now any multiple of $(p, q)$ will be on the line. Any uch multiple $(x, y)$ will satisfy the equation $$ q x - p y = 0 $$

hence this is the equation of the line you're looking for.