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I am working on estimation of a covariance matrix and I know that the matrix is Toeplitz. The desired matrix should not produce negative eigenvalues at all. However, sometime my estimation leads to a Toeplitz matrix which has few negative eigenvalues along with many positives. So far I do not know why the matrix has negative eigenvalues; I am sure my estimation procedure is not appropriate and I am trying to improve it.

So, I am trying to understand if there is any specific properties of Toeplitz matrix that lead to negative eigenvalues? Or is there a way to predict whether a Toeplitz matrix would produce negative eigenvalue without computing the eigenvalues.

Specifically, consider an $n \times n$ toeplitz matrix $T_n = [t_k,_j; k,j = 0, 1,... ,n − 1]$. My question is can we predict the presence of negative eigenvalues of $T_n$ form $t_k,_j$? There are limiting relations between Fourier Transform of $t_k,_j$ and the eigenvalues and hence I was thinking of the possibility. A reference for the limiting relation can be found in:http://ee.stanford.edu/~gray/toeplitz.pdf

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  • $\begingroup$ Not that I know, apart from very specific cases (such as being diagonally dominant). $\endgroup$
    – Federico Poloni
    May 16, 2016 at 9:19
  • $\begingroup$ Using Schmidt-Spitzer theorem, you can figure out on what curves the eigenvalues accumulate as the size of the matrix increases... $\endgroup$
    – Per Alexandersson
    May 18, 2016 at 13:28

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