linear involution
Definition. Let $V$ be a vector space^{}. A linear involution is a linear operator $L:V\to V$ such that ${L}^{2}$ is the identity operator on $V$. An equivalent^{} definition is that a linear involution is a linear operator that equals 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 ${\u2102}^{n}$, 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 involution^{} operators and projection operators.
Theorem 2. Let $L$ and $P$ be linear operators on a
vector space $V$ over a field of characteristic^{} not 2, and let $I$ be the identity operator on $V$.
If $L$ is an involution then
the operators $\frac{1}{2}\left(I\pm L\right)$
are projection operators.
Conversely, if $P$ is a projection operator, then
the operators $\pm (2PI)$ are involutions.
Involutions have important application in expressing hermitianorthogonal 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 skewsymmetric operator permutable with it, i.e.
$$L=\overline{L}={L}^{t},{L}^{2}=I,S=\overline{S}={S}^{t},LS=SL.$$ 
Title  linear involution 

Canonical name  LinearInvolution 
Date of creation  20130322 13:34:37 
Last modified on  20130322 13:34:37 
Owner  matte (1858) 
Last modified by  matte (1858) 
Numerical id  14 
Author  matte (1858) 
Entry type  Definition 
Classification  msc 15A21 
Synonym  involution 
Related topic  Projection 
Related topic  AntiIdempotent 