So I have got the scalar product of two vectors defined in this way: $$(x,y)_c = x_1y_1+cx_2y_2$$ where $x = (x1,x2)$ and $y=(y1,y2)$. Now I need to find the set of tranformation matrices that preserves the scalar product defined above. I know that for the standard scalar product between two vectors the set of orthogonal matrices ${R_n}$ s.t. $R^T R = I$ preserves the scalar products: $$(Rp, Rq) = (p,q) \ \forall p,q$$
How do I find the equivalent transformation matrices for the scalar product above?
Let $C : c_{1,1} = 1, c_{2,2} = c, c_{1,2}=c_{2,1} = 0$. You want $(ACx)^TAy = (Cx)^Ty$, that is: $x^TCA^TAy = (Cx)^Ty$, then once again $A^TA=I$.
Edit: Note that you can extend that to arbitrary diagonal matrices $C$. So you can literally have $x \cdot_c y = \sum_{i=0}^{n}{c_{i,i}x_iy_i}$ in arbitrary dimensions and the requirement will still be $A^TA=I$ as seen above.