## Table of Contents

**Table of Contents**

Preface

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

Index

Reviews

“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.