Eigenvalue contour lines of Kac-Murdock-Szegő matrices with a complex parameter
This paper deals with the eigenvalues of the Toeplitz matrix \[K_n(\rho) =[\rho^{|j-k|}]_{j,k=1}^n=\left[\begin{array}{ccccc}1& \rho &\rho^2&\ldots &\rho^{n-1}\\ \rho &1 &\rho&\ldots &\rho^{n-2}\\ \rho^2& \rho& 1&\ldots &\rho^{n-3}\\ \vdots &\vdots &\vdots &\ddots &\vdots \\ \rho^{n-1}& \rho^{n-2} &\rho^{n-3}&\ldots &1\end{array}\right]\] with \(n\geq 3\) and \(\rho\in\mathbb{C}\). The main result is Theorem 4.1, that is proved using some specific properties of Chebyshev polynomials. Numerical results and illustrative examples support the theoretical findings.
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