Special classes of matrices
The study of many pure and applied mathematics often reduce to the
study of some special matrix sets. For example,
many optimization problems reduce to the study of nonnegative matrix
theory problems and there are many interesting unsolved
problems even for zero-one matrices (e.g., see [BeK], [BR], [KiK],
[LSOD], [NR]).
Other pure mathematics subjects such
as dynamical systems, matrix $K$-theory and
classification of special types of $C*$-algebras, etc. also involve
the study of nonnegative matrices
(e.g., see [Bla], [Dix], [KK], [Ky], [LW], [Mur] and their references).
Furthermore, there have been considerable interest in studying the convex sets
of positive semi-definite matrices, correlation matrices,
doubly nonnegative matrices, completely
positive matrices, co-positive matrices, etc.
(see [CV], [DJ], [DJLo], [GPW], [Lo], [LT]). In fact, the
study of completely positive maps on matrix spaces in the context of
$C^*$-algebras also reduces to the study of some
subcones of the cone of positive
semi-definite matrices (e.g., see [BPS], [C], [JKS],
[JOSV], [KK], [K], [LaS], [LoS], [PH]
and their references).
Therefore, it is worthwhile to continue to develop more
techniques to deal with these types of problems and applying the
results to different areas.
REFERENCES
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R. Bhat, V. Pati and V.S. Sunder, On some convex sets and
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B. Blackadar, K-Theory for Operator Algebras, MSRI publications no. 5,
Springer-Verlag, New York, 1986.
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R.A. Brualdi and H.J. Ryser, Combinatorial Matrix Theory, Cambridge University
Press, New York, 1991.
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M.D. Choi, Completely positive linear maps on complex matrices,
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J. Drew C.R. Johnson ,
The no long odd cycle theorem for
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J. Drew, C.R. Johnson and R. Loewy,
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C.R. Johnson, D.D.
Olesky, D.P. Stanford
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C.R. Johnson, M.K.
Kerr and D.P. Stanford,
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C.K. Li , D.P. Stanford, , D.D. Olesky
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Constant Row and Column Sums, Linear and Mutilinear Algebra, to appear.
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C.K. Li , and H. Woerdeman, Special classes of
positive and completely positive maps, Linear Algebra Appl., to appear.
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semi-definite matrices, Math. Ann. 253 (1980), 227-232.
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R. Loewy and H. Schneider,
Positive operators on the n-dimensional ice cream cone,
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J.A. Poluikis and R.D. Hill, Completely positive and
Hermitian-preserving linear transformations,
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