\( \newcommand{\cat}[1] {\mathrm{#1}} \newcommand{\catobj}[1] {\operatorname{Obj}(\mathrm{#1})} \newcommand{\cathom}[1] {\operatorname{Hom}_{\cat{#1}}} \newcommand{\multiBetaReduction}[0] {\twoheadrightarrow_{\beta}} \newcommand{\betaReduction}[0] {\rightarrow_{\beta}} \newcommand{\betaEq}[0] {=_{\beta}} \newcommand{\string}[1] {\texttt{"}\mathtt{#1}\texttt{"}} \newcommand{\symbolq}[1] {\texttt{`}\mathtt{#1}\texttt{'}} \newcommand{\groupMul}[1] { \cdot_{\small{#1}}} \newcommand{\groupAdd}[1] { +_{\small{#1}}} \newcommand{\inv}[1] {#1^{-1} } \newcommand{\bm}[1] { \boldsymbol{#1} } \newcommand{\qed} { {\scriptstyle \Box} } \require{physics} \require{ams} \require{mathtools} \)

Eigenvectors of projection and reflection matrices

Let \( P \) and \( R \) be the projection and reflection matrices for a vector \( a \). \( P \) has eigenvalues \(1\) and \(0\), while \( R \) has eigenvalues \(1\) and \(-1\). We can also say [what about the eigenvectors \( P \) and \( R \)?].