PlanetMath (more info)
 Math for the people, by the people.
Encyclopedia | Requests | Forums | Docs | Wiki | Random | RSS  
Login
create new user
name:
pass:
forget your password?
Main Menu
sections
Encyclopædia
Papers
Books
Expositions

meta
Requests (200)
Orphanage
Unclass'd
Unproven (490)
Corrections (6)
Classification

talkback
Polls
Forums
Feedback
Bug Reports

downloads
Snapshots
PM Book

information
News
Docs
Wiki
ChangeLog
TODO List
Legalese
About
Owner confidence rating: High Entry average rating: No information on entry rating
[parent] eigenvalues of an involution (Proof)

Proof. For the first claim suppose $ \lambda$ is an eigenvalue corresponding to an eigenvector $ x$ of $ A$. That is, $ Ax = \lambda x$. Then $ A^2x = \lambda Ax$, so $ x=\lambda^2x$. As an eigenvector, $ x$ is non-zero, and $ \lambda = \pm 1$. Now property (1) follows since the determinant is the product of the eigenvalues. For property (2), suppose that $ A-\lambda I = -\lambda A(A-1/\lambda I)$, where $ A$ and $ \lambda$ are as above. Taking the determinant of both sides, and using part (1), and the properties of the determinant, yields

$\displaystyle \det (A-\lambda I) = \pm \lambda^n \det(A-\frac{1}{\lambda} I).$
Property (2) follows. $ \Box$



"eigenvalues of an involution" is owned by Koro. [ owner history (1) ]
(view preamble)

View style:


This object's parent.
Log in to rate this entry.
(view current ratings)

Cross-references: sides, product, determinant, property, eigenvector, eigenvalue
There is 1 reference to this entry.

This is version 1 of eigenvalues of an involution, born on 2003-05-26.
Object id is 4301, canonical name is EigenvaluesOfAnInvolution.
Accessed 2031 times total.

Classification:
AMS MSC: 15A21 (Linear and multilinear algebra; matrix theory :: Canonical forms, reductions, classification)
Pending Errata and Addenda
None.
Discussion
Style: Expand: Order:
forum policy

No messages.

Interact
post | correct | update request | add example | add (any)