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I have raised this sort of question before but I think that now I've found a better term for the subject, one which might ring more bells for people - hence the repost. Hope you won't be too angry with me.

I am interested in eigenvector localization for deterministic matrices. There is a whole body of work on the random matrix setup but I am interested in bounding the ratio between eigenvector coordinates of certain fixed and messy matrices. Any results out there that you know of pertaining to this? (I know the old masters' work on positive/nonnegative matrices but need to handle further cases).

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some pointers on eigenvector localization, mainly in the context of Perron-Frobenius theory (which in view of your earlier post seems to be what you are looking for):

Principal eigenvectors of irregular graphs

Perron-frobenius theory for a generalized eigenproblem

Bounds for eigenvalues and eigenvectors of a nonnegative matrix which involve a measure of irreducibility

A remark on Minc's maximal eigenvector bound for positive matrices

Einschließung des positiven eigenvektors einer nichtnegativen, irreduziblen matrix

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Almost - I am trying to extend the results to the mixed-sign case. (Actually, it's slightly more complicated than that - I have a monotone matrix whose smallest eigenvector I am interested in; all the Perron-Frobenius theory applies to the inverse but the inverse is difficult to compute explicitly. That's why I keep looking for a way to obtain similar results for the original matrix) –  Felix Goldberg Sep 20 '12 at 15:52
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