Let $M$ be a $n \times n$ AR(1) matrix whose $(i,j)$-th entry is
$$M_{ij} = \rho^{|i-j|}$$
with $0 < \rho < 1$. Is there an explicit formula to compute the largest eigenvalue of $M$?
2026-02-22 23:34:28.1771803268
The largest eigenvalue of AR(1) matrix
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It seems that you have a Kac–Murdock–Szegö (KMS) matrix:
Ulf Grenander, Gabor Szegö, Toeplitz forms and their applications, University of California Press, Berkeley and Los Angeles, 1958.
William F. Trench, Asymptotic distribution of the spectra of a class of generalized Kac–Murdock–Szegö matrices, Linear Algebra and its Applications, Volume 294, Issues 1–3, 15 June 1999, pages 181-192.
Information on the eigenvalues can be found on page 182 of Trench's paper.