The * equicorrelation matrix* is the matrix where the entries on the diagonal are all equal to and all off-diagonal entries are equal to some parameter which lies in . If we were to write out the matrix, it would look something like this:

Alternatively, we can write it as , where is the column vector with all entries being 1 and is the identity matrix.

Here are some useful properties of the equicorrelation matrix:

- It is a Toeplitz matrix, and hence has the properties that all Toeplitz matrices has (see e.g. this link).
- It has two eigenvalues. The first eigenvalue is , with associated eigenvector . The second eigenvalue is , with associated eigenvectors , where the entries of areThis can be verified directly by doing the matrix multiplication.
- . This is because the determinant of a square matrix is equal to the product of its eigenvalues.
- is positive definite if and only if . A sketch of the proof can be found in the answer here. It boils down to proving some properties of the determinant expression in the previous point.
- . This can be verified directly by matrix multiplication. It can also be derived using the Sherman-Morrison formula.

Hi, there should be a typo in the inverse of Sigma, the `{\bf 11}^T` should be in the numerator, not in the denominator.

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Thanks for spotting that! I’ve amended the formula.

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What about the covariance matrix version of this correlation matrix, i.e. with diagonal element \sigma^2_i and off-diagonal value \rho\sigma_i\sigma_j ? Can we deduce some of its properties? Thanks!!

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If E is the equicorrelation matrix and S is the diagonal matrix with the \sigma_i’s on the diagonal, then the covariance matrix is simply SES. With this formula we should be able to derive analogs of the properties above.

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