What is an involutive operator

Question

Please help me in understanding this:

I have to find the eigen values of an involutive operator. So what exactly is an involutive operator? I mean I need one example for an involutive operator.

Note: I know that an operator A with property $ A^2 $ = 1 is involutive operator so does that mean $\begin{bmatrix} 0 &1 \\ 1 & 0 \end{bmatrix} $ and $\begin{bmatrix} 1 &0 \\ 0 & -1 \end{bmatrix} $ are involutive operators???

Answer 1

Suppose $A$ is a linear involutive operator $A: V \longrightarrow V$, $v \in V$ an eigenvector of $A$ ($v \neq 0$), $\lambda$ an eigenvalue such that $Av = \lambda v$.

Then $v = A^2v = A(\lambda v) = \lambda(Av) = \lambda^2 v$

hence $\lambda^2 = 1$. This implies that $\lambda = \pm1$.

Answer 2

An involution is an operator that is its own inverse, i.e. $\mathbf{A} = \mathbf{A}^{-1}$, see here. This implies that $\mathbf{A}^2=\mathbf{I}$, therefore your two matrices are in fact examples of involutions.