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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.

Source

Introduction to Linear Algebra. Strang, p291.