linear involutionLinearInvolution2003-04-20 17:04:292007-09-29 09:23:40Definition% this is the default PlanetMath preamble. as your knowledge
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\newcommand{\sZ}[0]{\mathbb{Z}}{\bf Definition.}
Let $V$ be a vector space.
A \emph{ 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.
{\bf 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 $\sC^n$, and $A$ is a $n\times n$ complex matrix, then we have that:
\begin{enumerate}
\item $\det A = \pm 1$.
\item The characteristic polynomial of $A$, $p(\lambda) = \det( A-\lambda I)$,
is a reciprocal polynomial, i.e.,
$$ p(\lambda) = \pm \lambda^n p(1/\lambda).$$
\end{enumerate}
(\PMlinkname{proof.}{EigenvaluesOfAnInvolution})
The next theorem gives a correspondence between involution
operators and projection operators.
{\bf 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 $\frac{1}{2}\big(I\pm L\big)$
are projection operators.
Conversely, if $P$ is a projection operator, then
the 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
$$H=Le^{iS},$$
being $L$ a real symmetric involution operator and $S$ a real skew-symmetric operator permutable with it, i.e.
$$L=\overline{L}=L^t, \qquad L^2=I, \qquad S=\overline{S}=-S^t, \qquad LS=SL.$$