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For a v -set X , W 23 ( v ) is a ( v 2 ) × ( v 3 ) inclusion matrix where rows and columns are indexed by pairs and triples of X , respectively, and for row T and column K , W 23 ( v ) ( T , K ) = 1 if T ⊆ K and zero otherwise. In this paper, we classify the basis elements of the null Z ( W 23 ( v ) ) , derived from the Gaussian elimination on W 23 ( v ) (called standard basis), into five classes. Then, we present a new algorithm to construct a ( 2 , 3 , v ) -halving for a feasible v , i.e. a nowhere zero T ( 2 , 3 , v ) -trade. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we investigate the properties of a special kind of periodic Jacobi matrices. We show that the solution of the inverse problem for periodic Jacobi matrices is unique if and only if the matrix is of that special kind. Moreover, we present an algorithm to construct the solution of that inverse problem and give some illustrative examples. Finally, we perform a stability analysis of the algorithm. [Copyright &y& Elsevier]

Abstract: This paper presents a connection between qualitative matrix theory and linear complementarity problems (LCPs). An LCP is said to be sign-solvable if the set of the sign patterns of the solutions is uniquely determined by the sign patterns of the given coefficients. We provide a characterization for sign-solvable LCPs such that the coefficient matrix has nonzero diagonals, which can be tested in polynomial time. This characterization leads to an efficient combinatorial algorithm to find the sign pattern of a solution for these LCPs. The algorithm runs in time, where is the number of the nonzero coefficients. [Copyright &y& Elsevier]

Abstract: In the present paper we focus our research on calculating minors of weighing matrices of order n and weight n − k, denoted by W(n, n − k). We provide analytical determinant computations, counting techniques for specifying the existence of certain submatrices inside a W(n, n − k) and an algorithm for computing the (n − j)×(n − j) minors of a W(n, n − k), which is realized with the notion of symbolic manipulation. These results are valid of general n. The ideas presented in this work can be used as the fundamental basis, on which the calculation of minors of other weighing matrices, and in general of orthogonal matrices, can be developed. An application of the derived formulas to an interesting problem of Numerical Analysis, the growth problem, is also presented. [Copyright &y& Elsevier]

Abstract: A speed-up of a known O(n 3) algorithm computing the period of a periodic orbit in max–min algebra is presented. If the critical components (or the transitive closure A +) of the transition matrix A are known, the computational complexity of the algorithm is O(n 2). This is achieved by using only those coordinates of the orbit that are related to the critical components. On the other hand, no critical component can be omitted. As the critical components are pairwise disjoint, the new formula is helpful also in solving the converse problem: generating an orbit with a prescribed period. This is demonstrated by examples. All parts of the paper are connected with the fact that the periodic regime of an orbit is encoded in its critical coordinates. We show that the non-critical coordinates of the state vector after n −1 steps can be ignored, how to generate a known periodic regime by using a small number of coordinates of the state vector and that the difference between the defect of an orbit and the quasidefect of its critical coordinates is small. The final part deals with the situation when the matrix is reduced after the orbit has reached its periodic regime. [Copyright &y& Elsevier]

Abstract: Let be any field and let B a matrix of . Zaballa found necessary and sufficient conditions for the existence of a matrix with prescribed similarity class and such that . In an earlier paper [A. Borobia, R. Canogar, Constructing matrices with prescribed off-diagonal submatrix and invariant polynomials, Linear Algebra Appl. 424 (2007) 615–633] we obtained, for fields of characteristic different from 2, a finite step algorithm to construct A when it exists. In this short note we extend the algorithm to any field. [Copyright &y& Elsevier]