A lower bound for the nearest correlation matrix problem based on the circulant mean

This paper deals with the problem of finding the nearest correlation matrix to a given n x n symmetric matrix . We say that is a correlation matrix if it is symmetric positive semidefinite whose diagonal entries are equal to . The nearest correlation matrix to is the one that minimizes its distance to with relation to the Euclidean or Frobenius norm. The circulant mean of , denoted by , is the matrix obtained by averaging over the diagonals of , with the diagonals being extended to the length by a wrap-around. It turns out (due to a simple application of Jensen's trace inequality) that the distance from to its nearest correlation matrix will be a lower bound for the original distance. As the main result of this work, we compute the distance from to its nearest correlation matrix by relating it to the eigenvalues of . A motivation of the choice of the circulant mean is the following: we apply this lower bound and a related upper bound to the special case of the Laplacian matrix with Dirichlet boundary conditions (a three-band Toeplitz matrix, which is not circulant, that is, is not equal to its circulant mean). In this application we show that the distance to its nearest correlation matrix behaves asymptotically as with C-assymp approximate to 1.1165.