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{matrix} 0.5 & 0.5 \\ 0.5 & 0.5 \end{matrix}]

and

R = [\begin{matrix} 0 & 1 \\ 1 & 0 \end{matrix}] .

Note that

R = 2 P - 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.