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## linear algebra – largest eigenvalue of a symmetric Toeplitz matrix using Szegö’s theorem

Given a vector $v \in \mathbb{R}^w$which can be thought of as the kernel of an MA process, one can compute the symmetric Toeplitz matrix with entries:

$$ M_{i, j} = f(|ij|) = \sum_{h=0}^{h

This matrix is the covariance matrix of this stationary process. I am interested in the largest eigenvalue of $M$ at the limit where the size of $M$ goes to infinity.

Note that in the case where all entries of $v$ are positive, I get that the largest eigenvalue is

$$ (\sum_{h=0}^{h

I can also conjecture that in the general case, the solution is the square of the maximum of the spectral function:

$$ \max_{\omega \in \mathbb{R}} | \sum_h v_h e^{i\omega h}|^2 $$which seems to be an application of Szegö’s theorem.

Is there a way to compute this quantity analytically?

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