eigenvalues of an involution

Proof. For the first claim suppose λ is an eigenvalueMathworldPlanetmathPlanetmathPlanetmathPlanetmath corresponding to an eigenvectorMathworldPlanetmathPlanetmathPlanetmath x of A. That is, Ax=λx. Then A2x=λAx, so x=λ2x. As an eigenvector, x is non-zero, and λ=±1. Now property (1) follows since the determinantMathworldPlanetmath is the product of the eigenvalues. For property (2), suppose that A-λI=-λA(A-1/λI), where A and λ are as above. Taking the determinant of both sides, and using part (1), and the properties of the determinant, yields


Property (2) follows.

Title eigenvalues of an involution
Canonical name EigenvaluesOfAnInvolution
Date of creation 2013-03-22 13:38:57
Last modified on 2013-03-22 13:38:57
Owner Koro (127)
Last modified by Koro (127)
Numerical id 4
Author Koro (127)
Entry type Proof
Classification msc 15A21