| Version 5 |
Version 4 |
| {\bf Definition.} |
{\bf Definition.} |
| Let $V$ be a vector space. |
Let $V$ be a vector space. |
| A \emph{ linear involution} is a linear |
A \emph{ linear involution} is a linear |
| operator $L:V\to V$ such that $L^2$ is the identity operator on $V$. |
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 |
An equivalent definition is that a linear involution is a linear operator that |
| equals to it's own inverse. |
equals to it's own inverse. |
|
|
| {\bf Theorem 1.} Let $V$ be a vector space and let $A:V\to V$ be a linear involution. |
{\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, |
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: |
if $V$ is $\sC^n$, and $A$ is a $n\times n$ complex matrix, then we have that: |
| \begin{enumerate} |
\begin{enumerate} |
| \item $\det A = \pm 1$. |
\item $\det A = \pm 1$. |
| \item The characteristic polynomial of $A$, $p(\lambda) = \det( A-\lambda I)$, |
\item The characteristic polynomial of $A$, $p(\lambda) = \det( A-\lambda I)$, |
| is a reciprocal polynomial, i.e., |
is a reciprocal polynomial, i.e., |
| $$ p(\lambda) = \pm \lambda^n p(1/\lambda).$$ |
$$ p(\lambda) = \pm \lambda^n p(1/\lambda).$$ |
| \end{enumerate} |
\end{enumerate} |
| (\PMlinkname{proof.}{EigenvaluesOfAnInvolution}) |
(\PMlinkname{proof.}{EigenvaluesOfAnInvolution}) |
|
|
| The next theorem gives a correspondence between involution |
The next theorem gives a correspondence between involution |
| operators and projection operators. |
operators and projection operators. |
|
|
| {\bf Theorem 2.} Let $L$ and $P$ be linear operators on a |
{\bf Theorem 2.} Let $L$ and $P$ be linear operators on a |
| vector space $V$, and let $I$ be the identity operator on $V$. |
vector space $V$, and let $I$ be the identity operator on $V$. |
| If $L$ is an involution then |
If $L$ is an involution then |
| the operators $\frac{1}{2}\big(I\pm L\big)$ |
the operators $\frac{1}{2}\big(I\pm L\big)$ |
| are projection operators. |
are projection operators. |
| Conversely, if $P$ is a projection operator, then |
Conversely, if $P$ is a projection operator, then |
| the operators $\pm(2P-I)$ are involutions. |
the operators $\pm(2P-I)$ are involutions. |
|
|
| The next theorem is given as exercise IV.10.14 in \cite{pease}. |
The next theorem is given as exercise IV.10.14 in \cite{pease}. |
|
|
| {\bf Theorem 3.} Let $A$ be a complex $n\times n$ matrix. Then any two |
{\bf Theorem 3.} Let $A$ be a complex $n\times n$ matrix. Then any two |
| the the below conditions imply the third: |
the the below conditions imply the third: |
| \begin{enumerate} |
\begin{enumerate} |
| \item $A$ is a Hermitian matrix. |
\item $A$ is a Hermitian matrix. |
| \item $A$ is unitary matrix. |
\item $A$ is unitary matrix. |
| \item The mapping $A:\sC^n \to \sC^n$ is an involution. |
\item The mapping $A:\sC^n \to \sC^n$ is an involution. |
| \end{enumerate} |
\end{enumerate} |
|
|
| The proofs of theorems 2 and 3 are straightforward calculations. |
The proofs of theorems 2 and 3 are straightforward calculations. |
|
|
| \begin{thebibliography}{9} |
\begin{thebibliography}{9} |
| \bibitem {pease} M. C. Pease, |
\bibitem {pease} M. C. Pease, |
| \emph{Methods of Matrix Algebra}, |
\emph{Methods of Matrix Algebra}, |
| Academic Press, 1965 |
Academic Press, 1965 |
| \end{thebibliography} |
\end{thebibliography} |
| : |
: |
