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1 results

  • Approximating a Symmetric Matrix

  • R. A. Bailey, J. C. Gower
  • Journal:
    Psychometrika / Volume 55 / Issue 4 / December 1990
    Published online by Cambridge University Press:
    01 January 2025, pp. 665-675

  • We examine the least squares approximation C to a symmetric matrix B, when all diagonal elements get weight w relative to all nondiagonal elements. WhenB 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.