linear involution

Definition.Let $V$ be a vector space.A linear involution is a linearoperator $L:V\to V$ such that $L^2$ is the identity operator on $V$.An equivalent definition is that a linear involution is a linear operator thatequals its own inverse.

Theorem 1. Let $V$ be a vector space and let $A:V\to V$ be a linear involution.Then the eigenvalues of $A$ are $\pm 1$. Further,if $V$ is ${ℂ}^n$, and $A$ is a $n\times n$ complex matrix, then we have that:

1. $detA=\pm 1$.

2. The characteristic polynomial of $A$, $p(\lambda )=det(A-\lambda I)$,is a reciprocal polynomial, i.e.,

   $$p(\lambda )=\pm {\lambda }^{n}p(1/\lambda ).$$

(proof. (http://planetmath.org/EigenvaluesOfAnInvolution))

The next theorem gives a correspondence between involutionoperators and projection operators.

Theorem 2. Let $L$ and $P$ be linear operators on avector space $V$ over a field of characteristic not 2, and let $I$ be the identity operator on $V$.If $L$ is an involution thenthe operators $\frac{1}{2}\left(I\pm L\right)$are projection operators.Conversely, if $P$ is a projection operator, thenthe operators $\pm (2P-I)$ are involutions.

Involutions have important application in expressing hermitian-orthogonal operators, that is, $H^t=\overline{H}=H^{-1}$. In fact, it may be represented as

$$H=L{e}^{iS},$$

being $L$ a real symmetric involution operator and $S$ a real skew-symmetric operator permutable with it, i.e.

$$L=\overline{L}=L^t,L^2=I,S=\overline{S}=-S^t,LS=SL.$$