APPROXIMATING A SYMMETRICAL 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 greater-than-or-equal to 1/2, then the rank of C is never greater than p, and when w less-than-or-equal-to 1/2 then the rank of C is at least p. For the problem of approximating a given n x n matrix with a zero diagonal by a squared-distance matrix, it is shown that the stress 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.