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Nonnegative ranks, decompositions, and factorizations of nonnegative matrices. (English) Zbl 0784.15001
Let $$A$$ be an $$m \times n$$ nonnegative matrix with elements in an ordered field. The smallest nonnegative integer $$q$$ for which there exist $$q$$ nonnegative column vectors such that each column of $$A$$ can be expressed as a nonnegative linear combination of those vectors is called the nonnegative column rank, $$c-\text{rank}_ +A$$. The nonnegative row rank, $$r-\text{rank}_ +A$$, is defined in a similar manner. Two other nonnegative ranks are defined for $$A$$ and all four are proved to be equal.
The authors show that in calculating these rank it is sufficient to calculate the appropriate nonnegative rank of a bivariate probability matrix or a stochastic matrix associated with $$A$$. They prove that if the (ordinary) rank, rank $$A$$, does not exceed 2 then $$\text{rank} A= \text{rank}_ +A$$. Finally, they show that the nonnegative rank can be computed exactly over the reals by a finite algorithm. They pose the problem of determining whether or not the nonnegative ranks of a rational matrix, over the reals and rationals, respectively, are equal.

##### MSC:
 15A03 Vector spaces, linear dependence, rank, lineability 15B48 Positive matrices and their generalizations; cones of matrices 15A23 Factorization of matrices 15B51 Stochastic matrices
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##### References:
 [1] Berman, A.; Hershkowitz, D., Combinatorial results on completely positive matrices, Linear algebra appl., 95, 111-125, (1987) · Zbl 0623.05040 [2] Berman, A.; Plemmons, R., Nonnegative matrices in the social sciences, (1979), Academic New York [3] Breiman, L., The ∏ method for estimating multivariate functions from noisy data, Technometrics, 33, 125-143, (1991) · Zbl 0742.62037 [4] Cohen, P.J., Decision procedures for real and p-adic fields, Comm. pure appl. math., 22, 131-151, (1969) · Zbl 0167.01502 [5] Collins, G.E., Quantifier elimination for real closed fields by cylindrical algebraic decomposition, (), 515-532 [6] Eaves, B.C.; Rothblum, U.G., A theory on extending algorithms for parametric problems, Math. oper. res., 14, 502-533, (1989) · Zbl 0673.90081 [7] Gregory, D.A.; Pullman, N.J., Semiring rank: Boolean rank and nonnegative rank factorization, J. combin. inform. system sci., 3, 223-233, (1983) · Zbl 0622.15007 [8] Harshman, R.A., Foundations of the \scparafac procedure: models and conditions for an ‘explanatory’ multi-modal factor analysis, UCLA working paper in phonetics, 16, (1970) [9] Hayashi, C., An algorithm for the solution of \scparafac model and analysis of the databy the international Rice adaptation experiments, Bull. biometric soc. Japan, 3, 77-91, (1982) [10] Henry, L., Schémas de nuptialité: Déséquilibre des sexes et célibat, Population (Paris), 24, 457-486, (1969) [11] Henry, L., Schémas de nuptialité: Déséquilibre des sexes et âge au mariage, Population (Paris), 24, 1067-1122, (1969) [12] Henry, L., Nuptiality, Theoret. population biol., 3, 135-152, (1972) [13] Jacobson, N., Lectures in abstract algebra III, (1964), Van Nostrand Princeton, N.J [14] Lancaster, P.; Tismenetsky, M., The theory of matrices with applications, (1985), Academic Orlando, Fla · Zbl 0516.15018 [15] Levin, B., On calculating maximum rank-one underapproximations for positive arrays, () [16] Nisan, N., Lower bounds for non-commutative computation, Proceedings of the 23rd annual ACM symposium on theory of computing, 410-418, (1991) [17] Renegar, J., On the computational complexity and geometry of the first-order theory of the reals, parts I, II, III, J. symbolic logic, 13, 255-352, (1992) · Zbl 0798.68073 [18] Saboulin, M.de, Remarques sur le célibat engendré par LES déséquilibres de la répartition par sexe, Union internationale pour l’etude scientifique de la population, 22, (1985), Florence, Séance F. [19] Seidenberg, A., A new decision method for elementary algebra, Ann. of math., 60, 365-374, (1954) · Zbl 0056.01804 [20] Suppes, P.; Zanotti, M., When are probabilistic explanations possible?, Synthese, 48, 191-199, (1981) · Zbl 0476.03011 [21] Tarski, A., A decision method for elementary algebra and geometry, (1951), Univ. of California Press Berkeley · Zbl 0044.25102 [22] Yannakakis, M., Expressing combinatorial optimization problems by linear programs, Proceedings of the 20th annual ACM symposium on theory of computing, 223-228, (1988)
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