How would I go about finding a reflection of a vector $p = \begin{pmatrix}p_1 \\ p_2 \\ \vdots \\ p_n\end{pmatrix}$ about the line $x = \lambda d$ using a linear transformation? I need a matrix result such that $Ap = q$, where $ q = \begin{pmatrix}q_1 \\ q_2 \\ \vdots \\ q_n\end{pmatrix}$.
I am dumbfounded on where to start.
Cheers for your help!
Best Answer
Assume that $d$ is a unit vector representing the reflexive axis. Decompose $p$ into parallel component $p_t=(p\cdot d)d$ and perpendicular component $p_n=p-p_t$. Then the result vector $q$ is just to reverse the perpendicular component and thus $$q=p_t-p_n=2p_t-p=2(p\cdot d)d-p=(2d~d\cdot-I)p$$ which concludes that required linear operator $A=2d\otimes d^T-I$