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Subtotally positive and Monge matrices. (English) Zbl 1090.15016
Unfortunately there is not a common terminology in the theory of total positivity. Some authors, following the fundamental book by S. Karlin on this theory [Total positivity. Vol. I (1968; Zbl 0219.47030)], use the names of totally positive and strictly totally positive matrices for matrices which other authors call, respectively, totally nonnegative and totally positive. This paper follows the second terminology, and studies the so called \(k\)-subtotally positive matrices, which were called in Karlin’s book strictly totally positive matrices of order \(k\) (\(STP_k\) matrices) and which have been referred to with this notation by some other authors.
The author of the present paper calls relevant a submatrix \(A_1\) of the matrix \(A\) if the rows as well as columns of \(A_1\) are consecutive in \(A\) and either the first row or the first column of \(A_1\) is, respectively, the first row or column of \(A\). The determinants of these submatrices were called by M. Gasca and J. M. Peña [Linear Algebra Appl. 165, 25–44 (1992; Zbl 0749.15010) and papers by the same authors] row-initial or column-initial minors of the matrix. The present paper proves that only \(mn\) inequalities determine whether an \(m\times n\) matrix is \(k\)-subtotally positive, for every \(k\), \(1\leq k\leq \min (m,n)\). The author studies some other properties of subtotally positive matrices, among them completion problems of 2-subtotally positive matrices and their additive counterpart, anti-Monge matrices.
The paper lacks of some references to the other terminologies and to related results, as mentioned above. For example, in Section 1, in the reference to theorem A of M. Fiedler and T. L. Markham [Linear Algebra Appl. 306, No. 1–3, 87–102 (2000; Zbl 0954.15017)], it should have been added that this theorem had been previously proved in a different way by [M. Gasca and J. M. Pena [loc. cit., theorem 4.1].

MSC:
15B48 Positive matrices and their generalizations; cones of matrices
15B57 Hermitian, skew-Hermitian, and related matrices
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[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] S. Fallat, private communication.
[3] Fiedler, M.; Markham, T.L., Generalized totally positive matrices, Linear algebra appl., 306, 87-102, (2000) · Zbl 0954.15017
[4] Fiedler, M., Equilibrated anti-Monge matrices, Linear algebra appl., 335, 151-156, (2001) · Zbl 0983.15023
[5] Horn, R.A.; Johnson, C.R., Topics in matrix analysis, (1991), Cambridge University Press Cambridge · Zbl 0729.15001
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