, Volume 22, Issue 1, pp 379–398 | Cite as

Semipositive matrices and their semipositive cones



The semipositive cone of \(A\in \mathbb {R}^{m\times n}, K_A = \{x\ge 0\,:\, Ax\ge 0\}\), is considered mainly under the assumption that for some \(x\in K_A, Ax>0\), namely, that A is a semipositive matrix. The duality of \(K_A\) is studied and it is shown that \(K_A\) is a proper polyhedral cone. The relation among semipositivity cones of two matrices is examined via generalized inverse positivity. Perturbations and intervals of semipositive matrices are discussed. Connections with certain matrix classes pertinent to linear complementarity theory are also studied.


Semipositive matrix Proper cone Polyhedral cone Generalized inverse positivity Linear complementarity Singular M-matrix Q-matrix 

Mathematics Subject Classification

15A48 90C33 15A23 15A09 



The first author thanks the Office of Alumni and International Relations, IIT Madras and the Department of Mathematics and Statistics, WSU for partial financial support for his visit to WSU in the winter of 2015. The authors thank M.S. Gowda of UMBC for his comments and suggestions, especially concerning Theorem 3.3. They also thank T. Parthasarathy for discussions on the linear complementarity problem.


  1. 1.
    Ben-Israel, A., Greville, T.N.E.: Generalized Inverses: Theory and Applications, 2nd edn., CMS Books in Mathematics/Ouvrages de Mathmatiques de la SMC, 15. Springer, New York (2003)Google Scholar
  2. 2.
    Berman, A., Plemmons, R.J.: Eight types of matrix monotonicity. Linear Algebra Appl. 13, 115–123 (1976)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Blattner, J.W.: Bordered matrices. J. Soc. Ind. Appl. Math. 10, 528–536 (1962)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Berman, A., Plemmons, R.J.: Nonnegative Matrices in the Mathematical Sciences, Revised Reprint of the 1979 Original, Classics in Applied Mathematics, vol. 9. SIAM, Philadelphia (1994)Google Scholar
  5. 5.
    Collatz, L.: Functional Analysis and Numerical Mathematics, Translated from the German by Hansjrg Oser. Academic Press, New York (1966)Google Scholar
  6. 6.
    Cottle, R.W., Pang, J.-S., Stone, R.E.: The Linear Complementarity Problem, Corrected Reprint of the 1992 Original, Classics in Applied Mathematics, vol. 60. Society for Industrial and Applied Mathematics (SIAM), Philadelphia (2009)Google Scholar
  7. 7.
    Eagambaram, N., Mohan, S.R.: On some classes of linear complementarity problems with matrices of order \(n\) and rank \((n-1)\). Math. Oper. Res. 15, 243–257 (1990)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Eaves, B.C.: The linear complementarity problem. Manag. Sci. 17, 612–634 (1971)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Fiedler, M., Pták, V.: Some generalizations of positive definiteness and monotonicity. Numer. Math. 9, 163–172 (1966)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Gowda, M.S.: Applications of degree theory to linear complementarity problems. Math. Oper. Res. 18, 868–879 (1993)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Horn, R.A., Johnson, C.R.: Matrix Analysis. Cambridge University Press, Cambridge (1990)MATHGoogle Scholar
  12. 12.
    Johnson, C.R., Kerr, M.K., Stanford, D.P.: Semipositivity of matrices. Linear Multilinear Algebra 37, 265–271 (1994)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Mangasarian, O.L.: Characterizations of real matrices of monotone kind. SIAM Rev. 10, 439–441 (1968)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Mangasarian, O.L.: Nonlinear Programming, Corrected Reprint of the 1969 Original, Classics in Applied Mathematics, vol. 10. SIAM, Philadelphia (1994)Google Scholar
  15. 15.
    Olech, C., Parthasarathy, T., Ravindran, G.: Almost N-matrices and linear complementarity. Linear Algebra Appl. 145, 107–125 (1991)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Parthasarathy, T.: \(P\)-Matrices and \(N\)-Matrices. Lecture Notes in Mathematics, vol. 977. Springer, Berlin (1983)Google Scholar
  17. 17.
    Rohn, J.: Inverse-positive interval matrices. Z. Angew. Math. Mech. 67(5), T492–T493 (1987)MathSciNetMATHGoogle Scholar
  18. 18.
    Tsatsomeros, M.: Geometric mapping properties of semipositive matrices. Linear Algebra Appl. 498, 349–359 (2016)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Department of MathematicsIndian Institute of Technology MadrasChennaiIndia
  2. 2.Department of Mathematics and StatisticsWashington State UniversityPullmanUSA

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