linear involution

Definition. Let V be a vector spaceMathworldPlanetmath. A linear involution is a linear operator L:VV such that L2 is the identity operator on V. An equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmath definition is that a linear involution is a linear operator that equals its own inverseMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath.

Theorem 1. Let V be a vector space and let A:VV be a linear involution. Then the eigenvaluesMathworldPlanetmathPlanetmathPlanetmathPlanetmath of A are ±1. Further, if V is n, and A is a n×n complex matrix, then we have that:

  1. 1.


  2. 2.

    The characteristic polynomialMathworldPlanetmathPlanetmath of A, p(λ)=det(A-λI), is a reciprocal polynomial, i.e.,


(proof. (

The next theorem gives a correspondence between involutionPlanetmathPlanetmath operators and projection operators.

Theorem 2. Let L and P be linear operators on a vector space V over a field of characteristicPlanetmathPlanetmath not 2, and let I be the identity operator on V. If L is an involution then the operators 12(I±L) are projection operators. Conversely, if P is a projection operator, then the operators ±(2P-I) are involutions.

Involutions have important application in expressing hermitian-orthogonal operators, that is, Ht=H¯=H-1. In fact, it may be represented as


being L a real symmetricPlanetmathPlanetmathPlanetmathPlanetmath involution operator and S a real skew-symmetric operator permutable with it, i.e.

Title linear involution
Canonical name LinearInvolution
Date of creation 2013-03-22 13:34:37
Last modified on 2013-03-22 13:34:37
Owner matte (1858)
Last modified by matte (1858)
Numerical id 14
Author matte (1858)
Entry type Definition
Classification msc 15A21
Synonym involution
Related topic Projection
Related topic AntiIdempotent