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This accessible book provides an introduction to the analysis and design of dynamic multiagent networks. Such networks are of great interest in a wide range of areas in science and engineering, including: mobile sensor networks, distributed robotics such as formation flying and swarming, quantum networks, networked economics, biological synchronization, and social networks. Focusing on graph theoretic methods for the analysis and synthesis of dynamic multiagent networks, the book presents a powerful new formalism and set of tools for networked systems. The book's three sections look at foundations, multiagent networks, and networks as systems. The authors give an overview of important ideas from graph theory, followed by a detailed account of the agreement protocol and its various extensions, including the behavior of the protocol over undirected, directed, switching, and random networks. They cover topics such as formation control, coverage, distributed estimation, social networks, and games over networks. And they explore intriguing aspects of viewing networks as systems, by making these networks amenable to control-theoretic analysis and automatic synthesis, by monitoring their dynamic evolution, and by examining higher-order interaction models in terms of simplicial complexes and their applications. The book will interest graduate students working in systems and control, as well as in computer science and robotics. It will be a standard reference for researchers seeking a self-contained account of system-theoretic aspects of multiagent networks and their wide-ranging applications. This book has been adopted as a textbook at the following universities: University of Stuttgart, GermanyRoyal Institute of Technology, SwedenJohannes Kepler University, AustriaGeorgia Tech, USAUniversity of Washington, USAOhio University, USA

Let A be a complex diagonalizable matrix with two distinct nonzero eigenvalues λ and μ, the Yang-Baxter-like matrix equation AXA = XAX is reconsidered. We correct and improve the results in Shen et al. (2020) when λ² − λμ + μ² = 0. We also improve the results in Shen et al. (2020) when λ² − λμ + μ² ≠ 0. We obtain the explicit structure of the solutions X for the Yang-Baxter-like matrix equation AXA = XAX, which are diagonalizable. Finally, we improve other existing relevant conclusions.

Let A be a singular diagonalizable complex matrix with three distinct eigenvalues. We derive all explicit solutions X of the Yang-Baxter-like matrix equation AXA = XAX, by taking advantage of the Jordan form structure of A. The result generates the formula obtained in Chen et al. (2019) and M. Saeed Ibrahim Adam et al. (2019). We give examples to illustrate the validity of the results obtained in this paper.

Let A be a diagonalizable complex matrix. In this paper, we discuss finding solutions to the Yang–Baxter-like matrix equation AXA=XAX. We then present a concrete example to illustrate the validity of the results obtained.

Let be a square matrix satisfying . We solve the Yang-Baxter-like matrix equation to find some solutions, based on analysis of the characteristic polynomial of and its eigenvalues. We divide the problem into small cases so that we can find the solution easily. Finally, in order to illustrate the results, two numerical examples are presented.

For general complex or real 1-parameter matrix flows A ( t ) n , n and for static matrices A ∈ ℂ n , n alike, this paper considers ways to decompose matrix flows and single matrices globally via one constant matrix similarity C n , n as A ( t ) = C − 1 ⋅ diag( A 1 ( t ),..., A ℓ ( t )) ⋅ C or A = C − 1 ⋅diag( A 1 ,..., A ℓ ) ⋅ C with each diagonal block A k ( t ) or A k square and their number ℓ exceeding 1 if this is possible. The theory behind our proposed algorithm is elementary and uses the concept of invariant subspaces for the MATLAB eig computed ‘eigenvectors’ of one associated flow matrix B ( t a ) to find the coarsest simultaneous block structure for all flow matrices B ( t b ). The method works efficiently for all time-varying matrix flows A ( t ), be they real or complex, normal, with Jordan structures or repeated eigenvalues, differentiable, continuous, or discontinuous in t , and likewise for all fixed entry matrices A . Our intended aim is to discover unitarily diagonal-block decomposable flows as they originate in real-time from sensor given data for time-varying matrix problems that are unitarily invariant. Then, the complexities of their numerical treatments decrease by adopting ‘divide and conquer’ methods for their diagonal blocks. In the process, we discover and study k-normal fixed entry matrix classes that can be decomposed under unitary similarities into various k -dimensional block-diagonal forms.

A review of the relatively little-known matrix class, called coninvolutions, is given. The properties of these matrices are compared with those of the well studied involutory matrices or, briefly, involutions.

The current paper is dedicated to proving the many properties of falling numbers and Stirling numbers. Then, these properties are implemented to construct the recurrence relations of the sums of powers ... .

Quaternions are a number system that has become increasingly useful for representing the rotations of objects in three-dimensional space and has important applications in theoretical and applied mathematics, physics, computer science, and engineering. This is the first book to provide a systematic, accessible, and self-contained exposition of quaternion linear algebra. It features previously unpublished research results with complete proofs and many open problems at various levels, as well as more than 200 exercises to facilitate use by students and instructors. Applications presented in the book include numerical ranges, invariant semidefinite subspaces, differential equations with symmetries, and matrix equations.Designed for researchers and students across a variety of disciplines, the book can be read by anyone with a background in linear algebra, rudimentary complex analysis, and some multivariable calculus. Instructors will find it useful as a complementary text for undergraduate linear algebra courses or as a basis for a graduate course in linear algebra. The open problems can serve as research projects for undergraduates, topics for graduate students, or problems to be tackled by professional research mathematicians. The book is also an invaluable reference tool for researchers in fields where techniques based on quaternion analysis are used.

Perturbation bounds for eigenvalues of diagonalizable matrices are derived. Perturbation bounds for singular values of arbitrary matrices are also given. We generalize some existing results.