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.
Best Answer
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:
$$ X+RX=2PX \quad \Rightarrow \quad RX=2PX-X=(2P-I)X $$