Abstract
We examine the least squares approximationC to a symmetric matrixB, when all diagonal elements get weightw relative to all nondiagonal elements. WhenB has positivityp andC is constrained to be positive semi-definite, our main result states that, whenw≥1/2, then the rank ofC is never greater thanp, and whenw≤1/2 then the rank ofC is at leastp. For the problem of approximating a givenn×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 withw=(n+2)/4; the main result remains true. Other related problems and algorithmic consequences are briefly discussed.
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Bailey, R.A., Gower, J.C. Approximating a symmetric matrix. Psychometrika 55, 665–675 (1990). https://doi.org/10.1007/BF02294615
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DOI: https://doi.org/10.1007/BF02294615