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Spectral properties of Kac-Murdock-Szeg\"o matrices with a complex parameter
Numerical Analysis · Submitted 22 Apr 2018
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim For $\rho \gt 1$, we develop simple approximate formulas for the eigenvalues and pinpoint all eigenvalues' locations.
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The abstract, sentence by sentence
When $0\lt \rho \lt 1$, the Kac-Murdock-Szeg\"o matrix $K_n(\rho)=\left[\rho^{\lvert j-k \rvert}\right]_{j,k=1}^n$ is a Toeplitz correlation matrix with many applications and very well known spectral properties. We study the eigenvalues and eigenvectors of $K_n(\rho)$ for the general case where $\rho$ is complex, pointing out similarities and differences to the case $0\lt \rho \lt 1$. We then specialize our results to real $\rho$ with $\rho \gt 1$, emphasizing the continuity of the eigenvalues as functions of $\rho$. For $\rho \gt 1$, we develop simple approximate formulas for the eigenvalues and pinpoint all eigenvalues' locations. Our study starts from a certain polynomial whose zeros are connected to the eigenvalues by elementary formulas. We discuss relations of our results to earlier results of W. F. Trench.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | When $0\lt \rho \lt 1$, the Kac-Murdock-Szeg\"o matrix $K_n(\rho)=\left[\rho^{\lvert j-k \rvert}\right]_{j,k=1}^n$ is a Toeplitz correlation matrix with many applications and very well known spectral properties. | background 1.00 | 0.05 | 0.03 | asserted 1.00 |
| s1 | We study the eigenvalues and eigenvectors of $K_n(\rho)$ for the general case where $\rho$ is complex, pointing out similarities and differences to the case $0\lt \rho \lt 1$. | contribution 1.00 | 0.93 | 0.11 | asserted 0.97 |
| s2 | We then specialize our results to real $\rho$ with $\rho \gt 1$, emphasizing the continuity of the eigenvalues as functions of $\rho$. | contribution 0.92 | 0.94 | 0.11 | asserted 0.95 |
| s3 | For $\rho \gt 1$, we develop simple approximate formulas for the eigenvalues and pinpoint all eigenvalues' locations. | result 0.73 | 0.96 | 0.32 | asserted 1.00 |
| s4 | Our study starts from a certain polynomial whose zeros are connected to the eigenvalues by elementary formulas. | method 0.99 | 0.82 | 0.04 | asserted 1.00 |
| s5 | We discuss relations of our results to earlier results of W. F. Trench. | background 0.60 | 0.36 | 0.03 | asserted 0.67 |
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The paper
Original abstract
When $0\lt \rho \lt 1$, the Kac-Murdock-Szeg\"o matrix $K_n(\rho)=\left[\rho^{\lvert j-k \rvert}\right]_{j,k=1}^n$ is a Toeplitz correlation matrix with many applications and very well known spectral properties. We study the eigenvalues and eigenvectors of $K_n(\rho)$ for the general case where $\rho$ is complex, pointing out similarities and differences to the case $0\lt \rho \lt 1$. We then specialize our results to real $\rho$ with $\rho \gt 1$, emphasizing the continuity of the eigenvalues as functions of $\rho$. For $\rho \gt 1$, we develop simple approximate formulas for the eigenvalues and pinpoint all eigenvalues' locations. Our study starts from a certain polynomial whose zeros are connected to the eigenvalues by elementary formulas. We discuss relations of our results to earlier results of W. F. Trench.
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