The first-order autoregressive (AR(1)) correlation structure is important for applications in time series modeling and for repeated measures analysis. The AR(1) model provides a simple situations where measurements (on the same subject) that are closer in time are correlated more strongly than measurements recorded far apart. The AR(1) model uses
A colleague asked me an interesting question: Suppose you have a structured correlation matrix, such as a matrix that has a compound symmetric, banded, or an AR1(ρ) structure. If you generate a random correlation matrix that has the same eigenvalues as the structured matrix, does the random matrix have the
A previous article discusses covariance matrices that have the same set of eigenvalues. The set of eigenvalues is called the spectrum of the matrix. For symmetric matrices, the spectrum contains real numbers. For covariance matrices, which are positive semidefinite, the eigenvalues are nonnegative. It turns out that two symmetric matrices
