# 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 $\mathbb{C}^{n}$, and $A$ is a $n\times n$ complex matrix, then we have that:

1. 1.

$\det A=\pm 1$.

2. 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 $\frac{1}{2}\big{(}I\pm L\big{)}$ 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.

 $L=\overline{L}=L^{t},\qquad L^{2}=I,\qquad S=\overline{S}=-S^{t},\qquad LS=SL.$
Title linear involution LinearInvolution 2013-03-22 13:34:37 2013-03-22 13:34:37 matte (1858) matte (1858) 14 matte (1858) Definition msc 15A21 involution Projection AntiIdempotent