I have a question about the covariance matrix from a real-life study in particle physics. I am working on a non-linear fit of a theoretical prediction for a function T to N measurements D of this function carried out by experiment X. The experimental uncertainties of this measurement are provided in the form of a covariance matrix C, which enters the log-likelihood function as

$\chi^2 = \sum_{k,l} d_k C^{-1}_{kl} d_l$.

Here d_k=D_k-T_k is the difference between the theory and data for the point k,

and summation over k and l runs over all N data points. This is the usual definition of chi^2.

Each entry of C contains an uncorrelated statistical+systematic error s_i, as well as a correlated systematic uncertainty contributed by M unknown parameters common to the whole measurement. Thus in principle C should take the form

C = S + K, (1)

where S is a diagonal NxN matrix for the uncorrelated contribution, S = diag{s_1^2, ..., s_N^2}, and K is a matrix of rank M describing the correlated effects. The problem is that neither {s_i} nor M are known from the experimental publication for certain. The only available input in the whole matrix C. The question is: is it possible to find the likely values of {s_i} and M from the structure of C? What is the minimal value of M possible?

This question can be phrased as a linear algebra problem:

In a given representation for a real symmetric positive-definite NxN matrix C, express it in the form

C = S + K,

where S is a diagonal positive-definite matrix of rank N, and K is a semipositive-definite matrix of the _smallest_ possible rank M < N. Are the solutions for S and K unique? Outline an algorithm to find S and K.