  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.
