Advertisement

Mathematical Programming

, Volume 57, Issue 1–3, pp 439–444 | Cite as

AlmostP0-matrices and the classQ

  • Wallace C. Pye
Article

Abstract

This paper demonstrates that within the class of thosen × n real matrices, each of which has a negative determinant, nonnegative proper principal minors and inverse with at least one positive entry, the class ofQ-matrices coincides with the class of regular matrices. Each of these classes of matrices plays an important role in the theory of the linear complementarity problem. Lastly, analogous results are obtained for nonsingular matrices which possess only nonpositive principal minors.

Key words

Classes of matrices linear complementarity problem P0-matrix Q-matrix R-matrix andR0-matrix 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    M. Aganagić and R.W. Cottle, “A constructive characterization ofQ 0-matrices with nonnegative principal minors,”Mathematical Programming 37 (1987) 223–231.Google Scholar
  2. [2]
    M. Aganagić and R.W. Cottle, “A note onQ-matrices,”Mathematical Programming 16 (1979) 374–377.Google Scholar
  3. [3]
    R.W. Cottle, “The principal pivoting method of quadratic programming,” in: G.B. Dantzig and A.F. Veinott, Jr., eds.,Mathematics of the Decision Sciences, Part I (American Mathematical Society, Providence, RI, 1968) pp. 144–162.Google Scholar
  4. [4]
    R.W. Cottle, G.J. Habetler and C.E. Lemke, “On classes of copositive matrices,”Linear Algebra and Its Applications 3 (1970) 295–310.Google Scholar
  5. [5]
    R.D. Doverspike and C.E. Lemke, “A partial characterization of a class of matrices defined by solutions to the linear complementarity problem,”Mathematics of Operations Research 7 (1982) 272–294.Google Scholar
  6. [6]
    M. Fiedler and V. Pták, “Some generalizations of positive definiteness and monotonicity,”Numerische Mathematik 9 (1966) 163–172.Google Scholar
  7. [7]
    F.R. Grantmacher,The Theory of Matrices, Vol. 1 (Chelsea, New York, 1959).Google Scholar
  8. [8]
    M.W. Jeter and W.C. Pye, “Structure properties ofW-matrices,”Linear Algebra and Its Applications (1988) 219–229.Google Scholar
  9. [9]
    M.W. Jeter, W.C. Pye and C.E. Robinson, Jr., “Totally degenerate matrices,” in preparation.Google Scholar
  10. [10]
    S. Karamardian, “The complementarity problem,”Mathematical Programming 2 (1972) 107–129.Google Scholar
  11. [11]
    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
  12. [12]
    O.L. Mangasarian, “Characterization of linear complementarity problems as linear programs,”Mathematical Programming Study 7 (1978) 74–87.Google Scholar
  13. [13]
    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
  14. [14]
    J.D. Parsons, “Applications of principal pivoting,” in: H.W. Kuhn, ed.,Proceedings of the Princeton symposium on mathematical programming (Princeton University Press, Princeton, NJ, 1970) pp. 567–581.Google Scholar

Copyright information

© The Mathematical Programming Society, Inc. 1992

Authors and Affiliations

  • Wallace C. Pye
    • 1
  1. 1.Department of MathematicsUniversity of Southern MississippiHattiesburgUSA

Personalised recommendations