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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)

15B48 Positive matrices and their generalizations; cones of matrices
15B57 Hermitian, skew-Hermitian, and related matrices
15A04 Linear transformations, semilinear transformations
Full Text: DOI
[1] Burkard, R.E.; Klinz, B.; Rudolf, R., Perspectives of Monge properties in optimization, Discrete appl. math., 70, 95-161, (1996) · Zbl 0856.90091
[2] M. Fiedler, Remarks on Monge matrices, Mathematica Bohemica (To appear.) · Zbl 1003.15022
[3] A.J. Hoffman, On simple linear programming problems, in: Convexity, Proc. Symposia in: Pure Mathematics, AMS, Providence, RI, 1961 pp. 317-327
[4] Rudolf, R.; Woeginger, G.J., The cone of Monge matrices: extremal rays and applications, ZOR methods models oper. res., 42, 161-168, (1995) · Zbl 0843.90101
[5] Marcus, M.; Minc, H., A survey of matrix theory and matrix inequalities, (1964), Allyn and Bacon Boston · Zbl 0126.02404
[6] Vandergraft, J.S., Spectral properties of matrices which have invariant cones, SIAM J. appl. math., 16, 1208-1218, (1968) · Zbl 0186.05701
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