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I am looking for popular symmetric positive definite matrices with explicitly known eigenvalues (at least largest eigenvalue) arising as autocovariance matrices in time series (for example). In fact, any positive definite matrix is a covariance matrix.

For example, if we consider the equicorrelation matrix $(1-\rho)I+\rho J$ where $J$ is the matrix of all $1'$s, then we have an explicit expression for all the eigenvalues. What about the Toeplitz matrix $\Sigma$ where $\Sigma_{ij}=\rho^{|i-j|}$? I searched for this but it seems we do not know an explicit formula for the leading eigenvalue although some asymptotic results are known.

So, is the equicorrelation matrix the only matrix with well known eigenvalues among the autocovariance matrices arising in time series? I would be glad to know other examples.

kjetil b halvorsen
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Landon Carter
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    Start with a diagonal matrix with non-negative entries. Conjugate it by any orthogonal matrix. Done. See https://stats.stackexchange.com/questions/215497 for details. – whuber Apr 10 '22 at 12:22
  • Thanks @whuber for the comment. Unfortunately, I am in a reverse situation. This is for a research project. We have shown that something happens if the eigenvalues of the covariance matrix satisfy some properties. We are now trying to see if real time series examples satisfy such conditions. For example, I could validate quite easily that the equicorrelation matrix satisfies the desired property. But, to say anything about say AR models, I need an explicit formula for the eigenvalues which doesn't seem to exist. – Landon Carter Apr 11 '22 at 23:37
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    That sounds quite different from the question you have actually asked. If you need an explicit formula or the equivalent, you must begin by telling us what the autocorrelation function is. Please edit your post to make it more relevant to your problem. – whuber Apr 12 '22 at 11:23

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