FAQFAQ   SearchSearch   MemberlistMemberlist   UsergroupsUsergroups 
 ProfileProfile   PreferencesPreferences   Log in to check your private messagesLog in to check your private messages   Log inLog in 
Forum index » Science and Technology » Math
symmetry in eigenvectors
Post new topic   Reply to topic Page 1 of 1 [4 Posts] View previous topic :: View next topic
Author Message
JEMebius
science forum Guru Wannabe


Joined: 24 Mar 2005
Posts: 209

PostPosted: Mon Jul 17, 2006 1:46 pm    Post subject: Re: symmetry in eigenvectors- Reply with quote

Vector y=(-x2 ,x1 ,-x4 ,x3) is obtained from vector x=(x1, x2, x3, x4)
by a double 4D rotation over 90 deg in the 1-2 and the 3-4 planes.

Whereas it is true that the other three eigenvectors of A are orthogonal
to x, there is no a priori reason to expect that y =must= be one of
these three.

However, by a suitable 4D coordinate transformation one can arrange
things such in the new coordinates x'1, x'2, x'3, x'4 the vector x is
unchanged: x = (x'1, x'2, x'3, x'4), and one of the remaining three
eigenvectors becomes (-x'2, x'1, -x'4, x'3).
It is a worthwhile exercise to find out which 4x4 symmetric matrices
have (x1, x2, x3, x4) and (-x2, x1, -x4, x3) as two of their eigenvectors.

Ciao: Johan E. Mebius



Al wrote:

Quote:
Hi!
Maybe someone could help me with the following. It would be very
helpful!

Given a real symmetric 4x4 matrix A (it's a covariance matrix), and one
real eigenvector x=(x1,x2,x3,x4).
Is it now possible to show that y=(-x2,x1,-x4,x3) is an eigenvector of
A, too?

Thanks,
Al


Back to top
Chip Eastham
science forum Guru


Joined: 01 May 2005
Posts: 412

PostPosted: Mon Jul 17, 2006 1:32 pm    Post subject: Re: symmetry in eigenvectors Reply with quote

Al wrote:
Quote:
Hi!
Maybe someone could help me with the following. It would be very
helpful!

Given a real symmetric 4x4 matrix A (it's a covariance matrix), and one
real eigenvector x=(x1,x2,x3,x4).
Is it now possible to show that y=(-x2,x1,-x4,x3) is an eigenvector of
A, too?

No, it isn't. While y is orthogonal to x, it is not true that
just because A is real symmetric, and x is an eigenvector
that y must also be an eigenvector.

It is true that eigenvectors of real symmetric A for distinct
eigenvalues must be orthogonal.

Here's a way to construct counterexamples to your
underlying claim. Take x (normalized) and extend
it to an orthonormal basis {x, u, v, w} for R^4 such
that y is not a multiple of any single basis element.

Define A = x'x + 2 u'u + 3 v'v + 4 w'w.

Clearly A is real symmetric. Due to orthonormality
of the basis, the eigenvalues are shown to be 1,2,3,4
corresponding to resp. basis x,u,v,w of eigenvectors.

Since y is not a multiple of any of these, y is not an
eigenvector of A.

regards, chip
Back to top
A N Niel
science forum Guru


Joined: 28 Apr 2005
Posts: 475

PostPosted: Mon Jul 17, 2006 1:16 pm    Post subject: Re: symmetry in eigenvectors Reply with quote

In article <1153141791.978301.175850@s13g2000cwa.googlegroups.com>, Al
<alalcoolj@gmx.de> wrote:

Quote:
Hi!
Maybe someone could help me with the following. It would be very
helpful!

Given a real symmetric 4x4 matrix A (it's a covariance matrix), and one
real eigenvector x=(x1,x2,x3,x4).
Is it now possible to show that y=(-x2,x1,-x4,x3) is an eigenvector of
A, too?


No.
Say A is diagonal matrix 1,1,1,0. Then (1,0,1,0) is an eigenvector,
but (0,1,0,1) is not.
Back to top
Al1131
science forum beginner


Joined: 01 Mar 2006
Posts: 4

PostPosted: Mon Jul 17, 2006 1:09 pm    Post subject: symmetry in eigenvectors Reply with quote

Hi!
Maybe someone could help me with the following. It would be very
helpful!

Given a real symmetric 4x4 matrix A (it's a covariance matrix), and one
real eigenvector x=(x1,x2,x3,x4).
Is it now possible to show that y=(-x2,x1,-x4,x3) is an eigenvector of
A, too?

Thanks,
Al
Back to top
Google

Back to top
Display posts from previous:   
Post new topic   Reply to topic Page 1 of 1 [4 Posts] View previous topic :: View next topic
The time now is Sun Sep 23, 2018 12:19 pm | All times are GMT
Forum index » Science and Technology » Math
Jump to:  

Similar Topics
Topic Author Forum Replies Last Post
No new posts jordan decomposition and generalized eigenvectors Jeremy Watts Undergraduate 0 Tue Jul 18, 2006 6:49 pm
No new posts jordan decomposition and generalized eigenvectors Jeremy Watts num-analysis 3 Tue Jul 18, 2006 6:48 pm
No new posts Eigenvectors/eigenvalues for large symmetric positive sem... Jon2 num-analysis 3 Wed Jul 05, 2006 1:34 pm
No new posts Eigenvectors with large matrices Alan Woodland num-analysis 5 Wed Jul 05, 2006 10:32 am
No new posts Where Do The Laws Of Physics Come From? From symmetry an... dkomo Math 0 Thu Jun 22, 2006 12:02 am

Copyright © 2004-2005 DeniX Solutions SRL
Other DeniX Solutions sites: Electronics forum |  Medicine forum |  Unix/Linux blog |  Unix/Linux documentation |  Unix/Linux forums  |  send newsletters
 


Powered by phpBB © 2001, 2005 phpBB Group
[ Time: 0.0131s ][ Queries: 20 (0.0022s) ][ GZIP on - Debug on ]