2023, Hassuneh, Imad, Adm, Mohammad, Garloff, Jürgen

The class of square matrices of order n having a positive determinant and all their minors up to order n-1 nonpositive is considered. A characterization of these matrices based on the Cauchon algorithm is presented which provides an easy test for their recognition. Furthermore, it is shown that all matrices lying between two matrices of this class with respect to the checkerboard ordering are contained in this class, too.

2021, Garloff, Jürgen, Al-Saafin, Doaa, Adm, Mohammad

In this article, the collection of classes of matrices which possess the interval property presented in [J. Garloff, M. Adm, and J. Titi, A survey of classes of matrices possessing the interval property and related properties, Reliab. Comput., 22:1-14, 2016] is continued. That is, given an interval of matrices with respect to a certain partial order, it is desired to know whether a special property of the entire matrix interval can be inferred from some of its elements lying on the vertices of the matrix interval. The interval property of some matrix classes found in the literature is presented, and the interval property of further matrix classes including the ultrametric, the conditionally positive semidefinite, and the infinitely divisible matrices is given for the first time. For the inverse M-matrices, the cardinality of the required set of vertex matrices known so far is significantly reduced.

2020-04-03, Al-Saafin, Doaa, Garloff, Jürgen

Let A = [a_ij] be a real symmetric matrix. If f: (0,oo) --> [0,oo) is a Bernstein function, a sufficient condition for the matrix [f(a_ij)] to have only one positive eigenvalue is presented. By using this result, new results for a symmetric matrix with exactly one positive eigenvalue, e.g., properties of its Hadamard powers, are derived.

2020, Adm, Mohammad, Al Muhtaseb, Khawla, Ghani, Ayed Abedel, Garloff, Jürgen

Totally nonnegative matrices, i.e., matrices having all their minors nonnegative, and matrix intervals with respect to the checkerboard partial order are considered. It is proven that if the two bound matrices of such a matrix interval are totally nonnegative and satisfy certain conditions then all matrices from this interval are totally nonnegative and satisfy these conditions, too, hereby relaxing the nonsingularity condition in a former paper [M. Adm, J. Garloff, Intervals of totally nonnegative matrices, Linear Algebra Appl. 439 (2013), pp.3796-3806].

2021, Alseidi, Rola, Margaliot, Michael, Garloff, Jürgen

Positive systems play an important role in systems and control theory and have found many applications in multi-agent systems, neural networks, systems biology, and more. Positive systems map the nonnegative orthant to itself (and also the nonpositive orthant to itself). In other words, they map the set of vectors with zero sign variation to itself. In this note, discrete-time linear systems that map the set of vectors with up to k-1 sign variations to itself are introduced. For the special case k = 1 these reduce to discrete-time positive linear systems. Properties of these systems are analyzed using tools from the theory of sign-regular matrices. In particular, it is shown that almost every solution of such systems converges to the set of vectors with up to k-1 sign variations. It is also shown that these systems induce a positive dynamics of k-dimensional parallelotopes.

2021, Titi, Jihad, Garloff, Jürgen

Matrix methods for the computation of bounds for the range of a complex polynomial and its modulus over a rectangular region in the complex plane are presented. The approach relies on the expansion of the given polynomial into Bernstein polynomials. The results are extended to multivariate complex polynomials and rational functions.

2020, Alseidi, Rola, Garloff, Jürgen

In this paper, rectangular matrices whose minors of a given order have the same strict sign are considered and sufficient conditions for their recognition are presented. The results are extended to matrices whose minors of a given order have the same sign or are allowed to vanish. A matrix is called oscillatory if all its minors are nonnegative and there exists a positive integer k such that A^k has all its minors positive. As a generalization, a new type of matrices, called oscillatory of a specific order, is introduced and some of their properties are investigated.

2021, Adm, Mohammad, Garloff, Jürgen

The class of square matrices of order n having a negative determinant and all their minors up to order n-1 nonnegative is considered. A characterization of these matrices is presented which provides an easy test based on the Cauchon algorithm for their recognition. Furthermore, the maximum allowable perturbation of the entry in position (2,2) such that the perturbed matrix remains in this class is given. Finally, it is shown that all matrices lying between two matrices of this class with respect to the checkerboard ordering are contained in this class, too.

2021, Adm, Mohammad, Garloff, Jürgen, Titi, Jihad, Elgayar, Ali

Bounding the range of a sum of rational functions is an important task if, e.g., the global polynomial sum of ratios problem is solved by a branch abd bound algorithm. In this paper, bounding methods are discussed which rely on the expansion of a multivariate polynomial into Bernstein polynomials.

2020, Titi, Jihad, Garloff, Jürgen

The expansion of a given multivariate polynomial into Bernstein polynomials is considered. Matrix methods for the calculation of the Bernstein expansion of the product of two polynomials and of the Bernstein expansion of a polynomial from the expansion of one of its partial derivatives are provided which allow also a symbolic computation.