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 $M$?
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It seems that you have a Kac–Murdock–Szegö (KMS) matrix:
M. Kac, W. L. Murdock, G. Szegö, On the eigen-values of certain Hermitian forms, Journal of Rational Mechanics and Analysis, Volume 2, 1953.
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.
William F. Trench, Spectral distribution of generalized Kac–Murdock–Szegö matrices, Linear Algebra and its Applications, Volume 347, Issues 1–3, 15 May 2002, Pages 251-273. [PDF]
Information on the eigenvalues can be found on page 182 of Trench's 1999 paper.
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Rodrigo de Azevedo
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