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linear involution (Definition)

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. $ \det A = \pm 1$.
  2. The characteristic polynomial of $ A$, $ p(\lambda) = \det( A-\lambda I)$, is a reciprocal polynomial, i.e.,
    $\displaystyle p(\lambda) = \pm \lambda^n p(1/\lambda).$
(proof.)

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

$\displaystyle H=Le^{iS},$
being $ L$ a real symmetric involution operator and $ S$ a real skew-symmetric operator permutable with it, i.e.

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



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See Also: projection, anti-idempotent

Other names:  involution

Attachments:
eigenvalues of an involution (Proof) by Koro
example of linear involution (Example) by mathcam
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Cross-references: permutable, skew-symmetric, symmetric, real, application, characteristic, field, projection, operators, reciprocal polynomial, characteristic polynomial, matrix, complex, eigenvalues, inverse, equivalent, identity operator, linear operator, vector space
There are 6 references to this entry.

This is version 11 of linear involution, born on 2003-04-20, modified 2007-09-29.
Object id is 4197, canonical name is LinearInvolution.
Accessed 5178 times total.

Classification:
AMS MSC: 15A21 (Linear and multilinear algebra; matrix theory :: Canonical forms, reductions, classification)
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