Note: I'm using ";" to show in my matrices and vectors to represent the end of a row, e.g. [2; 4] is a column vector with 2 at the top and 4 at the bottom.

I know how to find the matrix of reflection through y = x in ℝ² (for which a = -b) by using the standard bases e1 = [1; 0], e2 = [0; 1] and got the matrix T = = [0 1; 1 0]

I tried exploring with y = -x (for which a = b) and got T = [0 -1; -1 0].

But I'm not seeing any pattern to a reflection across such a general case ax + by = 0.