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Equilibrated anti-Monge matrices. (English) Zbl 0983.15023
Some new properties of the so-called Monge/anti-Monge matrices [cf. A. J. Hoffman, Proc. Sympos. Pure Math. 7, 317-327 (1963; Zbl 0171.17801); R. E. Burkard, B. Klinz, and R. Rudolf, Discrete Appl. Math. 70, No. 2, 95-161 (1996; Zbl 0856.90091)] are investigated. The anti-Monge real \(n\times m\) matrices \(C=(c_{ik})\) \((c_{ik}+ c_{jl}\geq c_{il} +c_{jk}\) for all \(i<j\), \(k<l)\) are called equilibrated matrices if all their row sums as well as column sums are equal to zero. The theorem is proved by which every square equilibrated anti-Monge matrix is similar to a nonnegative matrix; the similarity matrix transforms non-negative vectors into monotone vectors \((u=(u_i)\), \(u_1\geq u_2\geq \cdots\geq u_k;\;\Sigma u_i=0)\). It is shown that such class of anti-Monge matrices is closed under multiplication.
Reviewer: A.A.Bogush (Minsk)

MSC:
15B48 Positive matrices and their generalizations; cones of matrices
15B57 Hermitian, skew-Hermitian, and related matrices
15A04 Linear transformations, semilinear transformations
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