Consider a Toeplitz matrix T which has both positive and negative eigenvalues. Can we prove that there exist two Toeplitz matrix T1 and T2 such that T1+T2=T and T1 has only one positive Eigenvalues and T2 has only negative Eigenvalues? Any reply would be appreciated. Regards
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$\begingroup$ Are you assuming your Toeplitz matrix is self-adjoint? $\endgroup$– Yemon ChoiCommented Aug 30, 2012 at 23:33
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$\begingroup$ The Toeplitz matrix considered is Hermitian matrix. Thank you for looking at my question. $\endgroup$– RantuCommented Aug 30, 2012 at 23:41
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1$\begingroup$ I think you need further conditions to make this interesting. You can take $T1 = T + \gamma I$, $T2 = - \gamma I$, where $\gamma$ is a large constant. This way, $T1$ will have only positive eigenvalues, and $T2$ will have negative ones (since you're shifting them by $\gamma$). $\endgroup$– comaCommented Aug 30, 2012 at 23:59
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$\begingroup$ True. Thank you for your comment. I assumed constant norm or unchanged diagonal condition need to be apply. Thanks a lot. $\endgroup$– RantuCommented Aug 31, 2012 at 0:07
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