Author

# Abraham Berman

Other affiliations: University of Wyoming, Centre de Recherches Mathématiques, University of Tennessee ...read more

Bio: Abraham Berman is an academic researcher from Technion – Israel Institute of Technology. The author has contributed to research in topic(s): Matrix (mathematics) & Nonnegative matrix. The author has an hindex of 29, co-authored 128 publication(s) receiving 9247 citation(s). Previous affiliations of Abraham Berman include University of Wyoming & Centre de Recherches Mathématiques.

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##### Papers

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01 Aug 1979

TL;DR: 1. Matrices which leave a cone invariant 2. Nonnegative matrices 3. Semigroups of non negative matrices 4. Symmetric nonnegativeMatrices 5. Generalized inverse- Positivity 6. M-matrices 7. Iterative methods for linear systems 8. Finite Markov Chains

Abstract: 1. Matrices which leave a cone invariant 2. Nonnegative matrices 3. Semigroups of nonnegative matrices 4. Symmetric nonnegative matrices 5. Generalized inverse- Positivity 6. M-matrices 7. Iterative methods for linear systems 8. Finite Markov Chains 9. Input-output analysis in economics 10. The Linear complementarity problem 11. Supplement 1979-1993 References Index.

6,318 citations

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01 Jan 1979

TL;DR: In this paper, the authors present a matrix theory background for positive control systems, and apply it to feedback control, including exponential nonnegative matrices and extended M-matrices.

Abstract: Convex Sets. Matrix Theory Background. Differential and Control System Preliminaries. Exponentially Nonnegative Matrices. Extended M-Matrices. Cone Reachability. Applications to Feedback Control. Controllability, Observability, and Realizability of Positive Control Systems. References. Index.

364 citations

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15 Apr 2003

TL;DR: In this article, the PSD Completion Problem Complete Positivity: Definition and Basic Properties Cones of Completely Positive Matrices Small Matrices complete positive matrix Small matrices complete positivity and the comparison matrix Completely positive graphs complete positive graph matrix complete positive graphs complete positive matrices of a given rank Complete positive matrix of the graph.

Abstract: Matrix Theoretic Background Positive Semidefinite Matrices Nonnegative Matrices and M-Matrices Schur Complements Graphs Convex Cones The PSD Completion Problem Complete Positivity: Definition and Basic Properties Cones of Completely Positive Matrices Small Matrices Complete Positivity and the Comparison Matrix Completely Positive Graphs Completely Positive Matrices Whose Graphs are Not Completely Positive Square Factorizations Functions of Completely Positive Matrices The CP Completion Problem CP Rank: Definition and Basic Results Completely Positive Matrices of a Given Rank Completely Positive Matrices of a Given Order When is the CP-Rank Equal to the Rank?

332 citations

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TL;DR: In this paper, various stability type conditions on a matrix A related to the consistency of the Lyapunov equation AD+DAt positive definite, where D is a positive diagonal matrix, are studied.

Abstract: We study various stability type conditions on a matrix A related to the consistency of the Lyapunov equation AD+DAt positive definite, where D is a positive diagonal matrix. Such problems arise in mathematical economics, in the study of time-invariant continuous-time systems and in the study of predator-prey systems. Using a theorem of the alternative, a characterization is given for all A satisfying the above equation. In addition, some necessary conditions for consistency and some related ideas are discussed. Finally, a method for constructing a solution D to the equation is given for matrices A satisfying certain conditions.

131 citations

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TL;DR: The concept of a regular splitting of a real nonsingular matrix, introduced by R. Varga, is used in iterative methods for solving linear systems as discussed by the authors, and it is shown that for a proper splitting, the spectral radius is less than l, if and only if the iteration converges to the best least squares approximate solution of the system.

Abstract: The concept of a regular splitting of a real nonsingular matrix, introduced by R. Varga, is used in iterative methods for solving linear systems. The purpose of this paper is to extend this concept to rectangular linear systems \[ ( * )\qquad Ax = b, \] in two directions. Firstly, this is accomplished by replacing $A^{ - 1} $ by $A^\dag $, the Moore–Penrose inverse of A, and, secondly, by considering matrices that leave a cone invariant.Let $A \in R^{m \times n} $. The splitting $A = M - N$ is called proper if $R(A) = R(M)$ and $N(A) = N(M)$. Such is the case when A and M are nonsingular. Consider the iteration \[ ( * * )\qquad x^{i + 1} = M^ \dag Nx^i + M^ \dag b. \] It is shown that for a proper splitting, $\rho (M^ \dag N)$ (the spectral radius of $M^ \dag N$) is less than l, if and only if the iteration (**) converges to $A^ \dag b$, the best least squares approximate solution of the system (*). This approach has the advantage of avoiding the normal system $A^T Ax = A^T b$ in solving (*). Necessary an...

127 citations

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01 Jan 1994

TL;DR: In this paper, the authors present a brief history of LMIs in control theory and discuss some of the standard problems involved in LMIs, such as linear matrix inequalities, linear differential inequalities, and matrix problems with analytic solutions.

Abstract: Preface 1. Introduction Overview A Brief History of LMIs in Control Theory Notes on the Style of the Book Origin of the Book 2. Some Standard Problems Involving LMIs. Linear Matrix Inequalities Some Standard Problems Ellipsoid Algorithm Interior-Point Methods Strict and Nonstrict LMIs Miscellaneous Results on Matrix Inequalities Some LMI Problems with Analytic Solutions 3. Some Matrix Problems. Minimizing Condition Number by Scaling Minimizing Condition Number of a Positive-Definite Matrix Minimizing Norm by Scaling Rescaling a Matrix Positive-Definite Matrix Completion Problems Quadratic Approximation of a Polytopic Norm Ellipsoidal Approximation 4. Linear Differential Inclusions. Differential Inclusions Some Specific LDIs Nonlinear System Analysis via LDIs 5. Analysis of LDIs: State Properties. Quadratic Stability Invariant Ellipsoids 6. Analysis of LDIs: Input/Output Properties. Input-to-State Properties State-to-Output Properties Input-to-Output Properties 7. State-Feedback Synthesis for LDIs. Static State-Feedback Controllers State Properties Input-to-State Properties State-to-Output Properties Input-to-Output Properties Observer-Based Controllers for Nonlinear Systems 8. Lure and Multiplier Methods. Analysis of Lure Systems Integral Quadratic Constraints Multipliers for Systems with Unknown Parameters 9. Systems with Multiplicative Noise. Analysis of Systems with Multiplicative Noise State-Feedback Synthesis 10. Miscellaneous Problems. Optimization over an Affine Family of Linear Systems Analysis of Systems with LTI Perturbations Positive Orthant Stabilizability Linear Systems with Delays Interpolation Problems The Inverse Problem of Optimal Control System Realization Problems Multi-Criterion LQG Nonconvex Multi-Criterion Quadratic Problems Notation List of Acronyms Bibliography Index.

10,744 citations

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TL;DR: A precise definition of the basic reproduction number, R0, is presented for a general compartmental disease transmission model based on a system of ordinary differential equations and it is shown that, if R0<1, then the disease free equilibrium is locally asymptotically stable; whereas if R 0>1,Then it is unstable.

Abstract: A precise definition of the basic reproduction number, Ro, is presented for a general compartmental disease transmission model based on a system of ordinary dierential equations. It is shown that, if Ro 1, then it is unstable. Thus,Ro is a threshold parameter for the model. An analysis of the local centre manifold yields a simple criterion for the existence and stability of super- and sub-threshold endemic equilibria for Ro near one. This criterion, together with the definition of Ro, is illustrated by treatment, multigroup, staged progression, multistrain and vectorhost models and can be applied to more complex models. The results are significant for disease control.

5,559 citations

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TL;DR: A Nyquist criterion is proved that uses the eigenvalues of the graph Laplacian matrix to determine the effect of the communication topology on formation stability, and a method for decentralized information exchange between vehicles is proposed.

Abstract: We consider the problem of cooperation among a collection of vehicles performing a shared task using intervehicle communication to coordinate their actions. Tools from algebraic graph theory prove useful in modeling the communication network and relating its topology to formation stability. We prove a Nyquist criterion that uses the eigenvalues of the graph Laplacian matrix to determine the effect of the communication topology on formation stability. We also propose a method for decentralized information exchange between vehicles. This approach realizes a dynamical system that supplies each vehicle with a common reference to be used for cooperative motion. We prove a separation principle that decomposes formation stability into two components: Stability of this is achieved information flow for the given graph and stability of an individual vehicle for the given controller. The information flow can thus be rendered highly robust to changes in the graph, enabling tight formation control despite limitations in intervehicle communication capability.

4,093 citations

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01 Jan 1974

TL;DR: In this paper, the Moore of the Moore-Penrose Inverse is described as a generalized inverse of a linear operator between Hilbert spaces, and a spectral theory for rectangular matrices is proposed.

Abstract: * Glossary of notation * Introduction * Preliminaries * Existence and Construction of Generalized Inverses * Linear Systems and Characterization of Generalized Inverses * Minimal Properties of Generalized Inverses * Spectral Generalized Inverses * Generalized Inverses of Partitioned Matrices * A Spectral Theory for Rectangular Matrices * Computational Aspects of Generalized Inverses * Miscellaneous Applications * Generalized Inverses of Linear Operators between Hilbert Spaces * Appendix A: The Moore of the Moore-Penrose Inverse * Bibliography * Subject Index * Author Index

3,834 citations