# Find a matrix for the linear transformation of reflection about a $\theta$ line using the matrix for projection

Projection matrix: $$\begin{pmatrix} \cos^2\theta & \cos\theta \sin\theta\\ \cos\theta \sin\theta & \sin^2\theta\\ \end{pmatrix}$$

Reflection matrix: $$\begin{pmatrix} 2\cos^2\theta -1 & 2\cos\theta sin\theta\\ 2\cos\theta \sin\theta & 2\sin^2\theta -1\\ \end{pmatrix}$$ which is equivalent to $$\begin{pmatrix} \cos2\theta & \sin2\theta\\ \sin2\theta & -\cos2\theta\\ \end{pmatrix}$$ I can arrive at the second form geometrically, but the problem asks that I do it with the projection matrix and vector addition. Any hints/tips?

It could also be that the problem is poorly worded, in which case a geometrical analysis is all that is necessary.

    "Find a formula for R based on P. Explain your arguments by drawing a
graph, using the rules of sums of vectors."


I drew the graph and was able to successfully explain it in terms of sums of vectors.

Hint:

use this fact: if $X$ is a point, $X'$ its projection on the axis and $X''$ the reflection of $X$, than $X'$ is the middle point of $XX''$.

so, if $P$ is the projection matrix and $R$ the reflection, we have:

$$X'=PX \qquad X''=RX$$ and, we write the fact that $X'$ is the midpoint as: $$X+X''=2X'$$

so:

$$X+RX=2PX \quad \Rightarrow \quad RX=2PX-X=(2P-I)X$$

• Yeah, it makes sense that the reflection is a projection onto the $\theta$ line, then another projection onto another $\theta$ line, but there's something missing. My textbook highlights it as $2P-I$, but it has no explanation for why the I pops up. – Wayfinder Mar 11 '16 at 16:33
• I've added to my answer. I hope it's useful. – Emilio Novati Mar 11 '16 at 16:41