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Complimentary Contributor Copy MATHEMATICS RESEARCH DEVELOPMENTS ADVANCES IN LINEAR ALGEBRA RESEARCH IVAN KYRCHEI EDITOR New York Complimentary Contributor Copy Copyright © 2015 by Nova Science Publishers, Inc. All rights reserved. No part of this book may be reproduced, stored in a retrieval system or transmitted in any form or by any means: electronic, electrostatic, magnetic, tape, mechanical photocopying, recording or otherwise without the written permission of the Publisher. For permission to use material from this book please contact us: [email protected] NOTICE TO THE READER The Publisher has taken reasonable care in the preparation of this book, but makes no expressed or implied warranty of any kind and assumes no responsibility for any errors or omissions. No liability is assumed for incidental or consequential damages in connection with or arising out of information contained in this book. The Publisher shall not be liable for any special, consequential, or exemplary damages resulting, in whole or in part, from the readers’ use of, or reliance upon, this material. Any parts of this book based on government reports are so indicated and copyright is claimed for those parts to the extent applicable to compilations of such works. Independent verification should be sought for any data, advice or recommendations contained in this book. In addition, no responsibility is assumed by the publisher for any injury and/or damage to persons or property arising from any methods, products, instructions, ideas or otherwise contained in this publication. This publication is designed to provide accurate and authoritative information with regard to the subject matter covered herein. It is sold with the clear understanding that the Publisher is not engaged in rendering legal or any other professional services. If legal or any other expert assistance is required, the services of a competent person should be sought. FROM A DECLARATION OF PARTICIPANTS JOINTLY ADOPTED BY A COMMITTEE OF THE AMERICAN BAR ASSOCIATION AND A COMMITTEE OF PUBLISHERS. Additional color graphics may be available in the e-book version of this book. LIBRARY OF CONGRESS CATALOGING-IN-PUBLICATION DATA Advances in linear algebra research / Ivan Kyrchei (National Academy of Sciences of Ukraine), editor. pages cm. -- (Mathematics research developments) Includes bibliographical references and index. ISBN: 978-1-63463-580-6 (eBook) 1. Algebras, Linear. I. Kyrchei, Ivan, editor. QA184.2.A38 2015 512'.5--dc23 2014043171 Published by Nova Science Publishers, Inc. † New York Complimentary Contributor Copy CONTENTS Preface vii Chapter 1 Minimization of Quadratic Forms and Generalized Inverses 1 Predrag S. Stanimirović, Dimitrios Pappas and Vasilios N. Katsikis Chapter 2 The Study of the Invariants of Homogeneous Matrix Polynomials 57 Using the Extended Hermite Equivalence εrh Grigoris I. Kalogeropoulos, Athanasios D. Karageorgos and Athanasios A. Pantelous Chapter 3 Cramer's Rule for Generalized Inverse Solutions 79 Ivan I. Kyrchei Chapter 4 Feedback Actions on Linear Systems over Von Neumann 133 Regular Rings Andrés Sáez-Schwedt Chapter 5 How to Characterize Properties of General Hermitian Quadratic 151 Matrix-Valued Functions by Rank and Inertia Yongge Tian Chapter 6 Introduction to the Theory of Triangular Matrices (Tables) 185 Roman Zatorsky Chapter 7 Recent Developments in Iterative Algorithms for Solving Linear 239 Matrix Equations Masoud Hajarian Chapter 8 Simultaneous Triangularization of a Pair of Matrices 287 over a Principal Ideal Domain with Quadratic Minimal Polynomials Volodymyr M. Prokip Chapter 9 Relation of Row-Column Determinants with Quasideterminants 299 of Matrices over a Quaternion Algebra Aleks Kleyn and Ivan I. Kyrchei Complimentary Contributor Copy vi Ivan Kyrchei Chapter 10 First Order Chemical Kinetics Matrices and Stability of O.D.E. 325 Systems Victor Martinez-Luaces About the Editor 345 Index 347 Complimentary Contributor Copy P REFACE This book presents original studies on the leading edge of linear algebra. Each chap- ter 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) descrip- tor (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 representa- tions 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 sin- gular 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. Chap- ter 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 paradetermi- nants 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 com- mon 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 non- commutative 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 ma- trix 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. Minimization of a quadratic form hx, T xi + hp, xi + a under constraints defined by a linear system is a common optimization problem. In Chapter 1, it is assumed that the Complimentary Contributor Copy viii Ivan Kyrchei operator T is symmetric positive definite or positive semidefinite. Several extensions to different sets of linear matrix constraints are investigated. Solutions of this problem may be given using the Moore-Penrose inverse and/or the Drazin inverse. In addition, several new classes of generalized inverses are defined minimizing the seminorm defined by the quadratic forms, depending on the matrix equation that is used as a constraint. A number of possibilities for further investigation are considered. In systems and control theory, Linear Time Invariant (LTI) descriptor (Differential- Algebraic) systems are intimately related to the matrix pencil theory. Actually, a large number of systems are reduced to the study of differential (difference) systems S (F, G) of the form: S (F, G): Fx˙(t) = Gx(t)(or the dual Fx = Gx˙(t)) , and m×n S (F, G): Fxk+1 = Gxk (or the dual Fxk = Gxk+1) , F, G ∈ C and their properties can be characterized by the homogeneous pencil sF −sGˆ . An essential problemin matrix pencil theory is the studyof invariantsof sF −sGˆ under the bilinear strict equivalence. This problem is equivalent to the study of complete Projective Equivalence (PE), EP , defined on the set Cr of complex homogeneous binary polynomials of fixed homogeneous degree r. For a f (s, sˆ) ∈ Cr, the study of invariants of the PE class EP is reduced to a study of invariants of matrices of the set Ck×2 (for k > 3 with all 2 ×2-minors non-zero) under the Extended Hermite Equivalence (EHE), Erh. In Chapter 2, the authors present a review of the most interesting properties of the PE and the EHE classes. Moreover, the appropriate projective transformation d ∈ RGL (1, C/R) is provided analytically ([1]). By a generalized inverse of a given matrix, the authors mean a matrix that exists for a larger class of matrices than the nonsingularmatrices, that has some of the properties of the