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2004 | 14 | 1 | 25-31
Tytuł artykułu

A linear programming based analysis of the CP-rank of completely positive matrices

Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
A real matrix A is said to be completely positive (CP) if it can be decomposed as A = BB^T, where the real matrix B has exclusively non-negative entries. Let k be the rank of A and Φ_k the least possible number of columns of the matrix B, the so-called completely positive rank (cp-rank) of A. The present work is devoted to a study of a general upper bound for the cp-rank of an arbitrary completely positive matrix A and its dependence on the ordinary rank k. This general upper bound of the cp-rank has been proved to be at most k(k + 1)/2. In a recent pioneering work of Barioli and Berman it was slightly reduced by one, which means that Φ_k ≤ k(k + 1)/2 - 1 holds for k ≥ 2. An alternative constructive proof of the same result is given in the present paper based on the properties of the simplex algorithm known from linear programming. Our proof illuminates complete positivity from a different point of view. Discussions concerning dual cones are not needed here. In addition to that, the proof is of constructive nature, i.e. starting from an arbitrary decomposition A = B_1B^T_1 (B_1 ≥ 0) a new decomposition A = B_2B^T_2 (B_2 ≥ 0) can be generated in a constructive manner, where the number of column vectors of B_2 does not exceed k(k + 1)/2 − 1. This algorithm is based mainly on the well-known techniques stemming from linear programming, where the pivot step of the simplex algorithm plays a key role.
Rocznik
Tom
14
Numer
1
Strony
25-31
Opis fizyczny
Daty
wydano
2004
otrzymano
2003-08-15
poprawiono
2003-11-10
Twórcy
autor
  • Department of Electrical and Information Engineering, University of Wuppertal, Rainer-Gruenter-Street 21, 42119 Wuppertal, Germany
  • Department of Electrical and Information Engineering, University of Wuppertal, Rainer-Gruenter-Street 21, 42119 Wuppertal, Germany
  • Department of Mathematics, University of Wuppertal, Gauß Street 20, 42119 Wuppertal, Germany
Bibliografia
  • Barioli F. and Berman A. (2003): The maximal cp-rank of rank completely positive matrices. - Linear Algebra Appl., Vol. 363, pp. 17-33.
  • Berman A. and Plemmons R. (1979): Nonnegative Matrices in the Mathematical Sciences. - New York: Academic Press.
  • Berman A. and Shaked-Monderer N. (2003): Completely Positive Matrices. - New York: World Scientific.
  • Berman A. (1993): Completely positive graphs, In: Combinatorial and Graph-Theoretical Problems in Linear Algebra (R.A. Brnaldi, S. Friedland and V. Klee, Eds.). - New York: Springer, pp. 229-233.
  • Drew J.H., Johnson C.R. and Loewy R. (1994): Completely positive matrices associated with M-matrices. - Linear and Multilinear Algebra, Vol. 37, No. 4, pp.303-310.
  • Drew J.H. and Johnson C.R. (1996): The no long odd cycle theorem for completely positive matrices, In: Random Discrete Structures (D. Aldous and R. Premantle, Eds.). - New York: Springer, pp. 103-115.
  • Golub G.H. and Van Loan Ch.F. (1989): Matrix Computations. - Baltimore: J. Hopkins University Press.
  • Glashoff K. and Gustafson S. (1983): Linear Optimization and Approximation. - New York: Springer.
  • Gray L.J. and Wilson D.G. (1980): Nonnegative factorization of positive semidefinite nonnegative matrices. - Linear Algebra Appl., Vol. 31.
  • Hall Jr. M. and Newman M. (1963): Copositive and completely positive quadratic forms. - Proc. Cambridge Philos. Soc., Vol. 59, pp. 329-339.
  • Hall Jr. M. (1967): Combinatorial Theory. - Lexington: Blaisdell Hanna J. and Laffey T.J. (1983): Nonnegative factorization of completely positive matrices. - Linear Algebra Appl., Vol. 55, pp. 1-9.
  • Karloff H. (1991): Linear Programming. - Boston: Birkh Kaykobad M. (1988): On nonnegative factorization of matrices. - Linear Algebra Appl., Vol. 96, pp. 27-33.
  • Kelly C. (1994): A test of the markovian model of DNA evolution. - Biometrics, Vol. 50, No. 3, pp. 653-664.
  • Kogan N. and Berman A. (1993): Characterization of completely positive graphs. - Discrete Math., Vol. 114, No. 1-3, pp. 297-304.
Typ dokumentu
Bibliografia
Identyfikatory
Identyfikator YADDA
bwmeta1.element.bwnjournal-article-amcv14i1p25bwm
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