Approximating a symmetric matrix

We examine the least squares approximation C to a symmetric matrix B, when all diagonal elements get weight w relative to all non-diagonal elements. When B has positivity p and C is constrained to be positive semi-definite, our main result states that, when w \geq 1/2, then the rank of C is never greater than p, and when w \leq 1/2 then the rank of C is at least p. For the problem of approximating a given n \times n matrix with zero diagonal by a squared-distance matrix, it is shown that the sstress criterion leads to a similar weighted least squares solution with w = (n+2)/4; the main result remains true. Other related problems and algorithmic consequences are briefly discussed.