Q-matrix

This article is about the notion used in the context of linear complementarity problems. For the one used in the context of Markov-chain, see Continuous-time Markov chain § Definition.

In mathematics, a Q-matrix is a square matrix whose associated linear complementarity problem LCP(M,q) has a solution for every vector q.

Properties

• M is a Q-matrix if there exists d > 0 such that LCP(M,0) and LCP(M,d) have a unique solution. ^{[1] [2]}
• Any P-matrix is a Q-matrix. Conversely, if a matrix is a Z-matrix and a Q-matrix, then it is also a P-matrix. ^[3]

See also

• P-matrix
• Z-matrix

References

1. Karamardian, S. (1976). "An existence theorem for the complementarity problem". Journal of Optimization Theory and Applications. 19 (2): 227–232. doi:10.1007/BF00934094. ISSN   0022-3239. S2CID   120505258.
2. Sivakumar, K. C.; Sushmitha, P.; Wendler, Megan (2020-05-17). "Karamardian Matrices: A Generalization of $Q$-Matrices". arXiv: 2005.08171 [math.OC].
3. Berman, Abraham. (1994). Nonnegative matrices in the mathematical sciences. Plemmons, Robert J. Philadelphia: Society for Industrial and Applied Mathematics. ISBN   0-89871-321-8. OCLC   31206205.

• Murty, Katta G. (January 1972). "On the number of solutions to the complementarity problem and spanning properties of complementary cones" (PDF). Linear Algebra and Its Applications. 5 (1): 65–108. doi:10.1016/0024-3795(72)90019-5. hdl: 2027.42/34188 .
• Aganagic, Muhamed; Cottle, Richard W. (December 1979). "A note on Q-matrices". Mathematical Programming. 16 (1): 374–377. doi:10.1007/BF01582122. S2CID   6384105.
• Pang, Jong-Shi (December 1979). "On Q-matrices". Mathematical Programming. 17 (1): 243–247. doi:10.1007/BF01588247. S2CID   209858727.
• Danao, R. A. (November 1994). "Q-matrices and boundedness of solutions to linear complementarity problems". Journal of Optimization Theory and Applications. 83 (2): 321–332. doi:10.1007/bf02190060. S2CID   121165848.