Math and science::Algebra
Eigenvectors of projection and reflection matricies
Let \( P \) and \( R \) be the projection and reflection matricies for a vector \( a \). \( P \) has eigenvalues \(1\) and \(0\), while \( R \) has eigenvalues \(1\) and \(-1\). We can also say \(P \) and \( R \) have the same eigenvectors.
Example
Let \( a = [1, 1] \), then
\[ P =
\begin{bmatrix}
0.5 & 0.5 \\
0.5 & 0.5
\end{bmatrix}
\]
and
\[
R =
\begin{bmatrix}
0 & 1 \\
1 & 0
\end{bmatrix}
.\]
Note that \( R = 2P - I \). The eigenvectors of both matricies are \( [1, 1] \)
and \( [1, -1] \). \( P \) has corresponding eigenvalues 1 and 0, while \( R \)
has corresponding eigenvalues 1 and -1.