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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 10 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 5043 times total.

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