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Let C(A)=A∘A−T{\mathcal{C}}\left(A)=A\circ {A}^{-T} be the combined matrix of an invertible matrix AA, where ∘\circ means the Hadamard product of matrices. In this work, we study the combined matrix of a nonsingular matrix, which is an HH-matrix whose comparison matrix is singular. In particular, we focus on C(A){\mathcal{C}}\left(A) when AA is diagonally equipotent, and we study whether C(A){\mathcal{C}}\left(A) is an HH-matrix and to which class it belongs. Moreover, we give some properties on the diagonal dominance of these matrices and on their comparison matrices.

In this paper we survey our work on preconditioners based on the Inverse Sherman-Morrison factorization. The most important theoretical results are also summarized and some numerical conclusions are provided.

The structural property of local reachability for positive two-dimensional (2-D) systems refers to single local states. The smallest number of steps needed to reach all local states of a system is the local reachability index of the system. This index may exceed the system dimension. Some authors have studied upper bounds on the local reachability index for specific positive 2-D systems and have suggested different upper bounds for any positive 2-D system. In this brief, the local reachability index for a special class of positive 2-D systems is characterized, and an upper bound for this index is derived. A comparison with previous results is also presented. Index Terms--Hurwitz products, influence digraph, local reachability index, nonnegative matrices, positive two-dimensional (2-D) systems, reachability.

An N-periodic forward-backward system defined by a collection of nonproper rational matrices is considered. In addition, an N-periodic collection of output feedbacks such that the closed-loop system is N-periodic and forward-backward is constructed. Block-ordered coprime factorizations of a rational matrix are used in the stabilization of that closed-loop system. Index Terms--Nonproper rational matrix, output-feedback, singular system.

Given a square matrix A, a Brauer's theorem [Brauer A., Limits for the characteristic roots of a matrix. IV. Applications to stochastic matrices, Duke Math. J., 1952, 19(1), 75-91] shows how to modify one single eigenvalue of A via a rank-one perturbation without changing any of the remaining eigenvalues. Older and newer results can be considered in the framework of the above theorem. In this paper, we present its application to stabilization of control systems, including the case when the system is noncontrollable. Other applications presented are related to the Jordan form of A and Wielandt's and Hotelling's deflations. An extension of the aforementioned Brauer's result, Rado's theorem, shows how to modify r eigenvalues of A at the same time via a rank- r perturbation without changing any of the remaining eigenvalues. The same results considered by blocks can be put into the block version framework of the above theorem. [ABSTRACT FROM AUTHOR]