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Jeremy Watts science forum Guru Wannabe
Joined: 24 Mar 2005
Posts: 239

Posted: Fri Jun 16, 2006 5:18 pm Post subject:
generalized eigenvector



the definition of a generalized eigenvector (not the same generalized
eigenvector of the generalized eigenvalue problem), is according to
'Schaum's Outlines' :
A vector Xm is a generalized vector if :
(A  lambdaI)^m Xm = 0
but,
(A  lambdaI)^(m1) Xm =/= 0
where Xm is an eigenvector of rank 'm', 'A' is a square matrix and 'lambda'
an associated eigenvalue.
But according to wikipedia, the definition is simply :
http://en.wikipedia.org/wiki/Generalized_eigenvector
why do these definitions differ?? 

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G.E. Ivey science forum Guru
Joined: 29 Apr 2005
Posts: 308

Posted: Fri Jun 16, 2006 6:31 pm Post subject:
Re: generalized eigenvector



Quote:  the definition of a generalized eigenvector (not the
same generalized
eigenvector of the generalized eigenvalue problem),
is according to
'Schaum's Outlines' :
A vector Xm is a generalized vector if :
(A  lambdaI)^m Xm = 0
but,
(A  lambdaI)^(m1) Xm =/= 0
where Xm is an eigenvector of rank 'm', 'A' is a
square matrix and 'lambda'
an associated eigenvalue.
But according to wikipedia, the definition is simply
:
http://en.wikipedia.org/wiki/Generalized_eigenvector
why do these definitions differ??
I don't see any difference! The definition you give essentially says that v is a generalized eigenvector if and only if (A lambdaI)^m v= v for some number m. The wikpedia reference talks about there being (A lambdaI) x_k= x_(k1) and, finally, (A lambdaI)^m x_(m1)= 0. Of course, if (A lambdaI)^m x= 0 then you can define the x_k as (A lambdaI)^k x. 


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