R help - Nov 2003 - real eigenvectors

Hello list,

Sorry, these questions are not directly linked to R.

If I consider an indefinte real matrix, I would like to know if the 
symmetry of the matrix is sufficient to say that their eigenvectors are real ?
And what is the conditions to ensure that eigenvectors are real in the case 
of an asymmetric matrix (if some conditions exist)?

Thanks in Advance,
St?phane DRAY
--------------------------------------------------------------------------------------------------

D?partement des Sciences Biologiques
Universit? de Montr?al, C.P. 6128, succursale centre-ville
Montr?al, Qu?bec H3C 3J7, Canada

Tel : 514 343 6111 poste 1233
E-mail : stephane.dray at umontreal.ca
--------------------------------------------------------------------------------------------------

Web                                          http://www.steph280.freesurf.fr/

<If I consider an indefinte real matrix ...>
I'm not familiar with the definition of an indefinite matrix.  

<I would like to know if the symmetry of the matrix is sufficient to say
that their eigenvectors are real ?>
(I assume you mean the eigenvalues are real...)
If A is a symmetric real valued matrix then the eignevalues of A are real.
There is a short proof of this due to Styan that can be found, among other
places, in Searle's 'Matrix Algebra Useful for Statistics' and
alluded to in
Anderson's Multivariate Analysis book in Chapter 11.

If you truly mean 'do the _eigenvectors_ have to be real?', then not if
you
allow A to operate on complex vectors.  If A is a real valued symmetric
matrix then it has a real valued eigenvalue, lambda, and a real valued
eigenvector, u which satisfy A * u = lambda * u.  If you allow A to operate
on complex vectors then i*u is a complex vector that satisfies A*(i*u)
lambda*(i*u).

< And what is the conditions to ensure that eigenvectors are real in the
case of an asymmetric matrix (if some conditions exist)?>
Again, I assume you mean eigenvalues are real, not eigenvectors.  I don't
think there is any (useful) condition in general.  The eigenvalues are real
if det(A-lambda*I) has real roots.  Think of the case of a 2x2 matrix where
you can solve this by the quadratic equation and consider some examples.

Bob


-----Original Message-----
From: Stephane DRAY [mailto:stephane.dray@umontreal.ca] 
Sent: Tuesday, November 04, 2003 1:14 PM
To: R help list
Subject: [R] real eigenvectors

Hello list,

Sorry, these questions are not directly linked to R.

If I consider an indefinte real matrix, I would like to know if the 
symmetry of the matrix is sufficient to say that their eigenvectors are real
?
And what is the conditions to ensure that eigenvectors are real in the case 
of an asymmetric matrix (if some conditions exist)?

Thanks in Advance,
Stéphane DRAY
----------------------------------------------------------------------------
---------------------- 

Département des Sciences Biologiques
Université de Montréal, C.P. 6128, succursale centre-ville
Montréal, Québec H3C 3J7, Canada

Tel : 514 343 6111 poste 1233
E-mail : stephane.dray@umontreal.ca
----------------------------------------------------------------------------
---------------------- 

Web
http://www.steph280.freesurf.fr/

______________________________________________
R-help@stat.math.ethz.ch mailing list
https://www.stat.math.ethz.ch/mailman/listinfo/r-help

	[[alternative HTML version deleted]]

On 4 Nov 2003 at 13:14, Stephane DRAY wrote:

That the matrix is symmetric is sufficient to guarantee real 
eigenvalues (proof is easy). I don't know about any general 
conditions for asymmetric matrices, and doubt there are. 

But in many structured situation you could be able to show similarity 
with a symmetric matrix, which suffices. Also note that symmetry 
doesn't need to be "visual" symmetry, it is enogh with symmetry
with respect to an inner product.

An example: Let A, B symmetric with B invertible . 
Then B^{-1}A has real eigenvalues, since it is similar to a symmetric 
matrix. 

Kjetil Halvorsen
> Hello list,
> 
> Sorry, these questions are not directly linked to R.
> 
> If I consider an indefinte real matrix, I would like to know if the 
> symmetry of the matrix is sufficient to say that their eigenvectors are
real ?
> And what is the conditions to ensure that eigenvectors are real in the case
> of an asymmetric matrix (if some conditions exist)?
> 
> Thanks in Advance,
> St?phane DRAY
>
--------------------------------------------------------------------------------------------------
> 
> D?partement des Sciences Biologiques
> Universit? de Montr?al, C.P. 6128, succursale centre-ville
> Montr?al, Qu?bec H3C 3J7, Canada
> 
> Tel : 514 343 6111 poste 1233
> E-mail : stephane.dray at umontreal.ca
>
--------------------------------------------------------------------------------------------------
> 
> Web                                         
http://www.steph280.freesurf.fr/
> 
> ______________________________________________
> R-help at stat.math.ethz.ch mailing list
> https://www.stat.math.ethz.ch/mailman/listinfo/r-help