Mathematical Programming

, Volume 53, Issue 1–3, pp 237–242 | Cite as

A note on a characterization of P-matrices

  • S. R. Mohan
  • R. Sridhar


In this note we show that the characterization results for P-matrices due to K.G. Murty and A. Tamir which state that a given square matrixM of ordern is a P-matrix if and only if the linear complementarity problem (q, M) has a unique solution for allq in a specified finite subsetГ of ℝ n depending onM are incorrect whenn > 3.

Key words

Linear complementarity problem P-matrix Q-matrix finite set characterization 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    J.T. Fredricksen, L.T. Watson and K.G. Murty, “A finite characterization of K-matrices in dimension less than four,”Mathematical Programming 35 (1986) 17–31.Google Scholar
  2. [2]
    A.W. Ingleton, “A problem in linear inequalities,”Proceedings of the London Mathematical Society 16 (1966) 519–536.Google Scholar
  3. [3]
    L.M. Kelly and L.T. Watson, “Q-matrices and spherical geometry,”Linear Algebra and its Applications 25 (1979) 175–189.Google Scholar
  4. [4]
    M. Kojima and R. Saigal, “On the number of solutions to a class of linear complementarity problems,”Mathematical Programming 17 (1979) 136–139.Google Scholar
  5. [5]
    K.G. Murty, “On the number of solutions to the complementarity problem and spanning properties of complementary cones,”Linear Algebra and its Applications 5 (1972) 65–108.Google Scholar
  6. [6]
    K.G. Murty, “On a characterization of P-matrices,”SIAM Journal on Applied Mathematics 20 (1971) 378–384.Google Scholar
  7. [7]
    K.G. Murty,Linear complementarity, Linear and Nonlinear Programming (Heldermann Verlag, Berlin, 1988).Google Scholar
  8. [8]
    H. Nikaido,Convex Structures and Economic Theory (Academic Press, New York, 1968).Google Scholar
  9. [9]
    R. Saigal, “On the class of complementary cones and Lemke's algorithm,”SIAM Journal on Applied Mathematics 23 (1972) 46–60.Google Scholar
  10. [10]
    H. Samelson, R.M. Thrall and O. Wesler, “A partition theorem for Euclideann-spaces,”Proceedings of the American Mathematical Society (1958) 805–807.Google Scholar
  11. [11]
    A Tamir, “On a characterization of P-matrices,”Mathematical Programming 4 (1973) 110–112.Google Scholar
  12. [12]
    A.W. Tucker, “Principal pivot transforms of square matrices,”SIAM Review 5 (1963) 305.Google Scholar

Copyright information

© The Mathematical Programming Society, Inc. 1992

Authors and Affiliations

  • S. R. Mohan
    • 1
  • R. Sridhar
    • 1
  1. 1.Indian Statistical InstituteNew DelhiIndia

Personalised recommendations