A 50 by 50 correlation matrix has 1 on the diagonal and rho = 0.3 everywhere off it. Give its eigenvalues with multiplicities, and state the smallest rho for which a matrix of this form with n = 50 is still positive semidefinite.

A 50 by 50 correlation matrix has 1 on the diagonal and rho = 0.3 everywhere off it. Give its eigenvalues with multiplicities, and state the smallest rho for which a matrix of this form with n = 50 is still positive semidefinite.

Approach: Write the matrix as (1 - rho) I + rho * 11^T, a rank one update of a multiple of the identity, and read the eigenvalues off the two invariant subspaces.

The eigenvalues are 1 + 49*0.3 = 15.7 once and 1 - 0.3 = 0.7 with multiplicity 49, and the equicorrelation matrix stays positive semidefinite for rho >= -1/49. Write C = (1 - rho) I + rho * 11^T. The all ones vector is an eigenvector of the rank one update 11^T with eigenvalue n = 50, giving 1 + (n-1)*rho = 15.7. Every vector orthogonal to 1 is killed by 11^T, so it is an eigenvector with eigenvalue 1 - rho = 0.7, and that subspace has dimension 49, which fixes the eigenvalue multiplicity. Both eigenvalues must be non-negative. The condition on the small eigenvalue caps rho at 1, and the condition on the large one floors it at -1/49, so rho lies in that closed range. The floor at -1/(n-1) is why a large set of assets cannot all be strongly negatively correlated with each other.

Follow-up: If you shrink a sample correlation matrix toward this equicorrelation target as (1-w) S + w C, what value of w guarantees the result is positive definite?

Key concepts: equicorrelation matrix, rank one update, positive semidefinite, eigenvalue multiplicity.