[R] real eigenvectors

kjetil@entelnet.bo kjetil at entelnet.bo
Tue Nov 4 23:31:41 CET 2003

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 

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,
> Stéphane DRAY
> -------------------------------------------------------------------------------------------------- 
> Département des Sciences Biologiques
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> Tel : 514 343 6111 poste 1233
> E-mail : stephane.dray at umontreal.ca
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