Let $R$ be the $2 \times 2$ “reflection” matrix which reflects points about the line $y = x$

Question

Let R be the 2 x 2 “reflection” matrix, which reflects points about the line y = x, (so (x, y) gets sent to (y, x)). Discuss what are the eigenvalues and eigenspaces of R.

Since R is a matrix that sends (x,y) to (y,x). that means that we have an equation that looks like this

R$\begin{pmatrix} x \\ y\end{pmatrix} = \begin{pmatrix} y \\ x\end{pmatrix}$

if we take the simplest case, of y = x, which is x = 1 and y = 1,

then the eigenvalues are 1 and the eigenvectors are (1,1)

are these the only eigenvalues/eigenvectors? or are there more?

Answer 1

Since$$R\begin{pmatrix}1\\1\end{pmatrix}=\begin{pmatrix}1\\1\end{pmatrix}\text{ and }R\begin{pmatrix}1\\-1\end{pmatrix}=\begin{pmatrix}-1\\1\end{pmatrix}=-\begin{pmatrix}1\\-1\end{pmatrix},$$the eigenvalues are $1$ and $-1$. The corresponding eigenspaces are $\mathbb{R}\begin{pmatrix}1\\1\end{pmatrix}$ and $\mathbb{R}\begin{pmatrix}1\\-1\end{pmatrix}$.

Answer 2

Hint: do not think algebraically, think geometrically instead. An eigenvector is a nonzero vector that keeps the same line after mapping. What vectors do that for a reflection in a line? Obviously, vectors that are on the line and vectors that are orthogonal to the line.