Q-matrix
{{About|the notion used in the context of linear complementarity problems|the one used in the context of Markov-chain|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.{{Cite journal|last=Karamardian|first=S.|date=1976|title=An existence theorem for the complementarity problem|journal=Journal of Optimization Theory and Applications|language=en|volume=19|issue=2|pages=227–232|doi=10.1007/BF00934094|s2cid=120505258|issn=0022-3239}}{{cite arXiv|last1=Sivakumar|first1=K. C.|last2=Sushmitha|first2=P.|last3=Wendler|first3=Megan|date=2020-05-17|title=Karamardian Matrices: A Generalization of $Q$-Matrices|class=math.OC|eprint=2005.08171}}
- 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.{{Cite book|last=Berman, Abraham.|title=Nonnegative matrices in the mathematical sciences|date=1994|publisher=Society for Industrial and Applied Mathematics|others=Plemmons, Robert J.|isbn=0-89871-321-8|location=Philadelphia|oclc=31206205}}
See also
References
{{Reflist}}
- {{cite journal|last1=Murty|first1=Katta G.|title=On the number of solutions to the complementarity problem and spanning properties of complementary cones|journal=Linear Algebra and Its Applications|date=January 1972|volume=5|issue=1|pages=65–108|doi=10.1016/0024-3795(72)90019-5|hdl=2027.42/34188|url=https://deepblue.lib.umich.edu/bitstream/2027.42/34188/1/0000477.pdf|hdl-access=free}}
- {{cite journal|last1=Aganagic|first1=Muhamed|last2=Cottle|first2=Richard W.|title=A note on Q-matrices|journal=Mathematical Programming|date=December 1979|volume=16|issue=1|pages=374–377|doi=10.1007/BF01582122|s2cid=6384105}}
- {{cite journal|last1=Pang|first1=Jong-Shi|title=On Q-matrices|journal=Mathematical Programming|date=December 1979|volume=17|issue=1|pages=243–247|doi=10.1007/BF01588247|s2cid=209858727}}
- {{cite journal|last1=Danao|first1=R. A.|title=Q-matrices and boundedness of solutions to linear complementarity problems|journal=Journal of Optimization Theory and Applications|date=November 1994|volume=83|issue=2|pages=321–332|doi=10.1007/bf02190060|s2cid=121165848}}
Category:Matrices (mathematics)
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