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 nondiagonal elements. When B has positivity p and C is constrained to be positive semi-definite, our main result states that, when w ≥1/2, then the rank of C is never greater than p , and when w ≤1/2 then the rank of C is at least p . For the problem of approximating a given n × n matrix with a 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.