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Generalization of real interval matrices to other fields
No full textAn interval matrix is a matrix whose entries are intervals in . This concept, which has been broadly studied, is generalized to other fields. Precisely, a rational interval matrix is defined to be a matrix whose entries are intervals in \Q. It is proved that a (real) interval matrix with the endpoints of all its entries in \Q contains a rank-one matrix if and only if it contains a rational rank-one matrix, and contains a matrix with rank smaller than if and only if it contains a rational matrix with rank smaller than ; from these results and from the analogous criterions for (real) inerval matrices, a criterion to see when a rational interval matrix contains a rank-one matrix and a criterion to see when it is full-rank, that is, all the matrices it contains are full-rank, are deduced immediately. Moreover, given a field and a matrix \al whose entries are subsets of , a criterion to find the maximal rank of a matrix contained in \al is described
Inequalities for permanents and permanental minors of row substochastic matrices
No full textIn this paper, some inequalities for permanents and permanental minors of row substochastic matrices are proved. The convexity of the permanent function on the interval between the identity matrix and an arbitrary row substochastic matrix is also proved. In addition, a conjecture about the permanent and permanental minors of square row substochastic matrices with fixed row and column sums is formulated
On Orthogonal Matrices with Zero Diagonal
No full textThis paper considers real orthogonal matrices whose diagonal entries are zero and off-diagonal entries nonzero, which are referred to as \OMZD(n). It is shown that there exists an \OMZD(n) if and only if , and that a symmetric \OMZD(n) exists if and only if is even and . Also, a construction of \OMZD(n) obtained from doubly regular tournaments is given. Finally, the results are applied to determine the minimum number of distinct eigenvalues of matrices associated with some families of graphs, and the related notion of orthogonal matrices with partially-zero diagonal is considered
On a Refined Operator Version of Young\u27s Inequality and Its Reverse
No full textIn this note, some refinements of Young\u27s inequality and its reverse for positive numbers are proved, and using these inequalities, some operator versions and Hilbert-Schmidt norm versions for matrices of these inequalities are obtained
Inequalities for sector matrices and positive linear maps
No full textAndo proved that if are positive definite, then for any positive linear map , it holds \begin{eqnarray*} \Phi(A\sharp_\lambda B)\le \Phi(A)\sharp_\lambda \Phi(B), \end{eqnarray*} where , , means the weighted geometric mean of . Using the recently defined geometric mean for accretive matrices, Ando\u27s result is extended to sector matrices. Some norm inequalities are considered as well
Decomposition of symplectic matrices into products of symplectic unipotent matrices of index 2
No full textIn this article, it is proved that every symplectic matrix can be decomposed into a product of three symplectic unipotent matrices of index 2, i.e., every complex matrix satisfying with is a product of three matrices satisfying and
Condensed Forms for Linear Port-Hamiltonian Descriptor Systems
No full textMotivated by the structure which arises in the port-Hamiltonian formulation of constraint dynamical systems, structure preserving condensed forms for skew-adjoint differential-algebraic equations (DAEs) are derived. Moreover, structure preserving condensed forms under constant rank assumptions for linear port-Hamiltonian differential-algebraic equations are developed. These condensed forms allow for the further analysis of the properties of port-Hamiltonian DAEs and to study, e.g., existence and uniqueness of solutions or to determine the index. It can be shown that under certain conditions for regular port-Hamiltonian DAEs the strangeness index is bounded by
Vector Cross Product Differential and Difference Equations in R^3 and in R^7
No full textThrough a matrix approach of the -fold vector cross product in and in , some vector cross product differential and difference equations are studied. Either the classical theory or convenient Drazin inverses, of elements belonging to the class of index matrices, are applied
SPN Graphs
No full textA simple graph G is an SPN graph if every copositive matrix having graph G is the sum of a positive semidefinite and nonnegative matrix. SPN graphs were introduced in [N. Shaked-Monderer. SPN graphs: When copositive = SPN. Linear Algebra Appl., 509:82{113, 2016.], where it was conjectured that the complete subdivision graph of K4 is an SPN graph. This conjecture is disproved, which in conjunction with results in the Shaked-Monderer paper show that a subdivision of K_4 is a SPN graph if and only if at most one edge is subdivided. It is conjectured that a graph is an SPN graph if and only if it does not have an F_5 minor, where F_5 is the fan on five vertices. To establish that the complete subdivision graph of K_4 is not an SPN graph, rank-1 completions are introduced and graphs that are rank-1 completable are characterized
