Advances in Linear Algebra Research


Ivan Kyrchei (Editor)
Pidstryhach Institute for Applied Problems of Mechanics and Mathematics, NAS of Ukraine, Lviv, Ukraine

Series: Mathematics Research Developments
BISAC: MAT002050

This book presents original studies on the leading edge of linear algebra. Each chapter has been carefully selected in an attempt to present substantial research results across a broad spectrum. The main goal of Chapter One is to define and investigate the restricted generalized inverses corresponding to minimization of constrained quadratic form. As stated in Chapter Two, in systems and control theory, Linear Time Invariant (LTI) descriptor (Differential-Algebraic) systems are intimately related to the matrix pencil theory. A review of the most interesting properties of the Projective Equivalence and the Extended Hermite Equivalence classes is presented in the chapter. New determinantal representations of generalized inverse matrices based on their limit representations are introduced in Chapter Three.

Using the obtained analogues of the adjoint matrix, Cramer’s rules for the least squares solution with the minimum norm and for the Drazin inverse solution of singular linear systems have been obtained in the chapter. In Chapter Four, a very interesting application of linear algebra of commutative rings to systems theory, is explored. Chapter Five gives a comprehensive investigation to behaviors of a general Hermitian quadratic matrix-valued function by using ranks and inertias of matrices. In Chapter Six, the theory of triangular matrices (tables) is introduced. The main “characters” of the chapter are special triangular tables (which will be called triangular matrices) and their functions paradeterminants and parapermanents.

The aim of Chapter Seven is to present the latest developments in iterative methods for solving linear matrix equations. The problems of existence of common eigenvectors and simultaneous triangularization of a pair of matrices over a principal ideal domain with quadratic minimal polynomials are investigated in Chapter Eight. Two approaches to define a noncommutative determinant (a determinant of a matrix with noncommutative elements) are considered in Chapter Nine. The last, Chapter 10, is an example of how the methods of linear algebra are used in natural sciences, particularly in chemistry. In this chapter, it is shown that in a First Order Chemical Kinetics Mechanisms matrix, all columns add to zero, all the diagonal elements are non-positive and all the other matrix entries are non-negative. As a result of this particular structure, the Gershgorin Circles Theorem can be applied to show that all the eigenvalues are negative or zero. (Imprint: Nova)



Table of Contents


Chapter 1. Minimization of Quadratic Forms and Generalized Inverses
Predrag S. Stanimirović, Dimitrios Pappas and Vasilios N. Katsikis (University of Niš, Faculty of Sciences and Mathematics, Niš, Serbia)

Chapter 2. The Study of the Invariants of Homogeneous Matrix Polynomials Using the Extended Hermite Equivalence εrh
Grigoris I. Kalogeropoulos, Athanasios D. Karageorgos and Athanasios A. Pantelous (Department of Mathematics, University of Athens, Greece)

Chapter 3. Cramer’s Rule for Generalized Inverse Solutions
Ivan I. Kyrchei (Pidstryhach Institute for Applied Problems of Mechanics and Mathematics, Lviv, Ukraine)

Chapter 4. Feedback Actions on Linear Systems over Von Neumann Regular Rings
Andrés Sáez-Schwedt (Departamento de Matemáticas, Universidad de León, Campus de Vegazana, León, Spain)

Chapter 5. How to Characterize Properties of General Hermitian Quadratic Matrix-Valued Functions by Rank and Inertia
Yongge Tian (CEMA, Central University of Finance and Economics, Beijing, China)

Chapter 6. Introduction to the Theory of Triangular Matrices (Tables)
Roman Zatorsky (Precarpathian Vasyl Stefanyk National University, Ivano-Frankivsk, Ukraine)

Chapter 7. Recent Developments in Iterative Algorithms for Solving Linear Matrix Equations
Masoud Hajarian (Department of Mathematics, Faculty of Mathematical Sciences, Shahid Beheshti University, General Campus, Evin, Tehran, Iran)

Chapter 8. Simultaneous Triangularization of a Pair of Matrices over a Principal Ideal Domain with Quadratic Minimal Polynomials
Volodymyr M. Prokip (Pidstryhach Institute for Applied Problems of Mechanics and Mathematics, Lviv, Ukraine)

Chapter 9. Relation of Row-Column Determinants with Quasideterminants of Matrices over a Quaternion Algebra
Aleks Kleyn and Ivan I. Kyrchei (American Mathematical Society and Ivan I. Kyrchei, Pidstryhach Institute for Applied Problems of Mechanics and Mathematics, NAS of Ukraine)

Chapter 10. First Order Chemical Kinetics Matrices and Stability of O.D.E. Systems
Victor Martinez-Luaces (Electrochemistry Engineering Multidisciplinary Research Group, Uruguay)

About the Editor



“This book presents some recent theoretical and computational results on Linear Algebra and Applications. The investigations developed in each one of its ten chapters have been written by one or more experts. They analyze subjects as quadratic forms, homogeneous matrix polynomials, linear control systems, Hermitian matrix-valued functions, triangular matrices, linear matrix equations, simultaneous triangularization over principal ideal domains and matrix differential equations, among others. Matrix Analysis Theory and Generalized Inverses are two extreme usefulness tools used to solve several of the proposed problems. In addition, settings such as the complex field or an arbitrary field, a ring or a quaternion algebra are the structures to work with. This interesting book is written in a very readable style and it is a very good contribution to the Linear Algebra Community and other interested readers.” -Nestor Thome, Professor, Department of Applied Mathematics of Polytechnical University of Valencia, Spain

“This is a worth-reading book about the recent developments in Linear Algebra and it includes contributions of fourteen authors from all over the world. The themes analyzed by the researchers in the ten chapters include quadratic optimization, matrix pencils, generalized inverses, matrix equations, maximal and minimal ranks and inertias, triangular matrices (tables) and their parafunctions, iterative methods, eigenvalues and eigenvectors, quasideterminants, regular rings and quaternions, among others. These developments have strong connections with other branches of mathematics like statistics, optimization, discrete mathematics and differential equations and they are related to important topics like fractals, graphs, power series, Markovian transitions and ODEs stability. Outside mathematics, potential applications to financial problems, electrical networks, filter design, chemical kinetics mechanisms and control theory, remark the importance of the topics considered. Finally, the inclusion of several open problems, numerical examples that clarify the theory and even a touch of humor in one of the footnotes, complete this interesting, enjoyable and easy readable book.”- Victor Martinez-Luaces, Profesor, Universidad de la República, Uruguay

“This book is a very interesting overview of recent topics of Linear Algebra and matrix Analysis. It includes topics such as Quadratic optimization, generalized inverses, Matrix Polynomials, Matrix functions, Iterative methods for solving matrix equations, simultaneous triangularization over a principal ideal domain, matrix differential equations and other very important topics. The book is quite easy to read and is written for postgraduate and/or PhD students, and researchers in the field of Linear Algebra. It includes theoretical investigations and many clarifying examples to support them, as well as numerical experiments presented. It can be used as a strong reference for scientific publications. Open problems on these topics are also presented and discussed.” -Dimitrios Pappas, Athens University of Economics and Business, Greece

“The book is devoted to advanced topics in Linear Algebra and its Application. It covers a broad range of topics. Some of them, for instance, paradeterminants and parapermanents, are not covered in the standard English books on the subject. Among the nice features of the book is the use of various tools of general algebra: fields, rings, quaternions, … The book is interesting and should be useful for experts and Ph.D. students in such areas of mathematics as PDE, control systems, graph theory, combinatorics, as well as in theoretical physics. It can also be of interest to engineers and researches working on the border between mathematics and chemistry, biology, or medicine.” – Professor Rostislav Grigorchuk, Mathematics Department of Texas A&M University

This book is written for a wide range of mathematicians, scientists dealing with linear algebra and its applications, and students which study linear algebra. It will be useful to all institutions where linear algebra and its applications are studied.

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