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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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Are you assuming your Toeplitz matrix is self-adjoint? – Yemon Choi Aug 30 at 23:33
The Toeplitz matrix considered is Hermitian matrix. Thank you for looking at my question. – Rantu Aug 30 at 23:41
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 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 $\gamma$). – coma Aug 30 at 23:59
True. Thank you for your comment. I assumed constant norm or unchanged diagonal condition need to be apply. Thanks a lot. – Rantu Aug 31 at 0:07

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