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.