I'm not familiar with the definition of an indefinite matrix.
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(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
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