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

plane geometry - Does the intersection of two moving curves sweep out a continuous line?

Hello dear visitor to our network We will offer you a solution to this question linear algebra – largest eigenvalue of a symmetric Toeplitz matrix using Szegö’s theorem ,and the answer will be typical through documented information sources, We welcome you and offer you new questions and answers, Many visitor are wondering about the answer to this question.

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?

we will offer you the solution to linear algebra – largest eigenvalue of a symmetric Toeplitz matrix using Szegö’s theorem question via our network which brings all the answers from multiple reliable sources.