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TRENDS IN FIELD THEORY RESEARCH MATHEMATICS RESEARCH DEVELOPMENTS HOT TOPICS IN LINEAR ALGEBRA No part of this digital document may be reproduced, stored in a retrieval system or transmitted in any fo.
MATHEMATICS RESEARCH DEVELOPMENTS HOT TOPICS IN LINEAR ALGEBRA No part of this digital document may be reproduced, stored in a retrieval system or transmitted in any form or by any means The publisher has taken reasonable care in the preparation of this digital document, 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 herein This digital document is sold with the clear understanding that the publisher is not engaged in rendering legal, medical or any other professional services MATHEMATICS RESEARCH DEVELOPMENTS Additional books and e-books in this series can be found on Nova’s website under the Series tab MATHEMATICS RESEARCH DEVELOPMENTS HOT TOPICS IN LINEAR ALGEBRA IVAN I KYRCHEI EDITOR Copyright © 2020 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 We have partnered with Copyright Clearance Center to make it easy for you to obtain permissions to reuse content from this publication Simply navigate to this publication’s page on Nova’s website and locate the “Get Permission” button below the title description This button is linked directly to the title’s permission page on copyright.com Alternatively, you can visit copyright.com and search by title, ISBN, or ISSN For further questions about using the service on copyright.com, please contact: Copyright Clearance Center Phone: +1-(978) 750-8400 Fax: +1-(978) 750-4470 E-mail: info@copyright.com 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 Names: Kyrchei, Ivan I., editor Title: Hot topics in linear algebra / Ivan Kyrchei, editor Identifiers: LCCN 2020015306 (print) | LCCN 2020015307 (ebook) | ISBN 9781536177701 (hardcover) | ISBN 9781536177718 (adobe pdf) Subjects: LCSH: Algebras, Linear Classification: LCC QA184.2 H68 2020 (print) | LCC QA184.2 (ebook) | DDC 512/.5 dc23 LC record available at https://lccn.loc.gov/2020015306 LC ebook record available at https://lccn.loc.gov/2020015307 Published by Nova Science Publishers, Inc † New York CONTENTS Preface vii Chapter Computing Generalized Inverses Using Gradient-Based Dynamical Systems Predrag S Stanimirović and Yimin Wei Chapter Cramer's Rules for Sylvester-Type Matrix Equations Ivan I Kyrchei Chapter BiCR Algorithm for Computing Generalized Bisymmetric Solutions of General Coupled Matrix Equations Masoud Hajarian 111 Chapter System of Mixed Generalized Sylvester-Type Quaternion Matrix Equations Abdur Rehman, Ivan I Kyrchei, Muhammad Akram, Ilyas Ali and Abdul Shakoor 137 Chapter Hessenberg Matrices: Properties and Some Applications Taras Goy and Roman Zatorsky 163 Chapter Equivalence of Polynomial Matrices over a Field Volodymyr M Prokip 205 45 vi Contents Chapter Matrices in Chemical Problems Modeled Using Directed Graphs and Multigraphs Victor Martinez-Luaces 233 Chapter Engaging Students in the Learning of Linear Algebra Marta G Caligaris, Georgina B Rodríguez and Lorena F Laugero 267 About the Editor 293 Index 295 P REFACE Linear algebra is the branch of mathematics concerning vector spaces and linear mappings between such spaces Systems of linear equations with several unknowns are naturally represented using the formalism of matrices and vectors So we arrive at the matrix algebra, etc Linear algebra is central to almost all areas of mathematics Many ideas and methods of linear algebra were generalized to abstract algebra Functional analysis studies the infinite-dimensional version of the theory of vector spaces Combined with calculus, linear algebra facilitates the solution of linear systems of differential equations Linear algebra is also used in most sciences and engineering areas, because it allows modeling many natural phenomena, and efficiently computing with such models ”Hot Topics in Linear Algebra” presents original studies in some areas of the leading edge of linear algebra Each article has been carefully selected in an attempt to present substantial research results across a broad spectrum Topics discussed herein include recent advances in analysis of various dynamical systems based on the Gradient Neural Network; Cramer’s rules for quaternion generalized Sylvester-type matrix equations; matrix algorithms for finding the generalized bisymmetric solution pair of general coupled Sylvester-type matrix equations; explicit solution formulas of some systems of mixed generalized Sylvester-type quaternion matrix equations; new approaches to studying the properties of Hessenberg matrices by using triangular tables and their functions; researching of polynomial matrices over a field with respect to semi-scalar equivalence; mathematical modeling problems in chemistry with applying mix- viii Ivan I Kyrchei ing problems which the associated MP-matrices; some visual apps, designed in Scilab, for the learning of different topics of Linear Algebra In Chapter 1, the dynamical systems and recurrent neural networks a apply as a powerful tool for solving many kinds of matrix algebra problems In particular, for computing generalized inverse matrices RNN models, that are dedicated to find zeros of equations or to minimize nonlinear functions and represent optimization networks, are used Convergence properties and exact solutions of considered models are investigated in this section as well In the following three chapters, matrix equations as one of more famous subjects of linear algebra are studying The well-known Cramer’s rule is an elegant formula for the solutions of a system of linear equations that has both theoretical and practical importances It is the consequence of the unique determinantal representation of inverse matrix by the adjoint matrix with the cofactors in the entries Is it possible to solve by Cramer’s rule the generalized Sylvester matrix equation AXC + BY D = E, (1) moreover when this equation has quaternionic coefficient matrices? Chapter gives the answer on this question In this chapter, Cramer’s rules for Eq (1) and for the quaternionic generalized Sylvester matrix equations with ∗- and ηHermicities are derived within the framework of the theory of noncommutative column-row determinants previously introduced by the author Algorithms of finding solutions are obtained in both cases with complex and quaternionic coefficient matrices In Chapter 3, the Hestenes-Stiefel (HS) version of biconjugate residual (BiCR) algorithm for finding the generalized bisymmetric solution pair (X, Y ) of the general coupled matrix equations f i=1 (Ai XBi + Ci Y Di) = M, g j=1 (Ej XFj + Gj Y Hj ) = N is established in a finite number of iterations in the absence of round-off errors Some necessary and sufficient conditions of constraint mixed type generalized Sylvester quaternion matrix equations A3 X = C3 , Y B3 = C4 , ZB4 = C5 , A4 ZB5 = C6 , A1 X − Y B1 = C1 , A2 X − ZB2 = C2 , Engaging Students in the Learning of Linear Algebra 283 Activity Given the following matrices: −1 𝐴2 = ( 𝐴1 = ( ) 0.5 ) −0.6 0.4 −0.25 𝐵2 = ( 1.5 𝐵1 = ( 0.8 ) −0.2 0.125 ) 0.25 0.5 ) 1.5 −1 𝐶2 = ( ) −12 −2 𝐶1 = ( a) Which is the result of A1 B1 and A2 B2 ? b) Can you find any relation between A1 and C1? And betweenA2 and C2? c) Calculate the eigenvalues and eigenvectors of each of the given matrices d) Can you draw any conclusion considering the previous answers? After performing this activity, students should deduce the following properties: If the eigenvalues of the matrix A are 1 , 2 , …, n , then the eigenvalues of the matrix A are 1 , 2 ,…, n If the eigenvalues of the matrix A are 1 , 2 , …, n , then the eigenvalues of the matrix A1 are 1/ 1 , 1/ 2 ,…, 1/ n The eigenvalues of A and A1 are equal The eigenvalues of A and A are equal Activity Given the following matrices: 𝐴1 = ( −2 −1 ) −4 𝐵1 = ( −1 ) a) Find the eigenvalues and eigenvectors of each matrix b) How many eigenvalues each matrix has? And eigenvectors? 284 Marta G Caligaris, Georgina B Rodríguez and Lorena F Laugero c) Study the linear dependence of the eigenvectors of each matrix d) Are you able to set a property based on the results obtained? Considering the obtained results, students should realize that the eigenvectors associated to a matrix are linearly independent, if they are associated to different eigenvalues Activity Given the points P(2, 1) and Q(-1, -2) in the canonical basis, find their coordinates referring to the basis A = {(1, 1), (-1, 1)} Observing the plot in the application, analyze the points´ position in the plane after the change of basis After the analysis, students should realize that the points not change their position on the plane A change of basis implies a change of the reference system, therefore the coordinates are different, but not the position Activity Consider the following linear transformations: 𝑇1 : ℝ2 → ℝ2 / 𝑇1 (𝑥, 𝑦) = (−𝑦, 𝑥) 𝑇2 : ℝ2 → ℝ2 / 𝑇2 (𝑥, 𝑦) = (2𝑥, 2𝑦) 𝑥 𝑇3 : ℝ2 → ℝ2 / 𝑇3 (𝑥, 𝑦) = ( )( ) 𝑦 −1 𝑥 𝑇4 : ℝ2 → ℝ2 / 𝑇4 (𝑥, 𝑦) = ( ) (𝑦) a) Apply each one of these transformations to the unit circumference centered in (0, 0) What can you say about the geometry of the image in each case? b) Can you find any relation between these linear transformations? c) How can you write a linear transformation in its matrix form if the associated matrix is expressed in the canonical basis? Engaging Students in the Learning of Linear Algebra 285 While performing this activity, students will discover that T and T4 have the same effect on the selected points (a 90° counter-clockwise rotation) The same happens with T2 and T3, as both produce an expansion with coefficient Therefore, students should conclude that a linear transformation can be expressed by different representations Although calculating the matrix of a linear transformation in the canonical basis is one of the simplest examples, this is a favorable situation to explain when a linear transformation may be expressed in this way It will be demonstrated in class that if V and W are vector spaces of dimensions n and m respectively (dim V = n and dim W = m), the linear transformation T: V →W admits a matrix representation as V is isomorph to ℝ𝑛 and W is isomorph to ℝ𝑚 Activity Consider the following linear transformations: 𝑇1 : ℝ2 → ℝ2 / 𝑇1 (𝑥, 𝑦) = ( 𝑥+𝑦 𝑥−𝑦 , ) 𝑥 𝑇2 : ℝ2 → ℝ2 / 𝑇2 (𝑥, 𝑦) = ( ) (𝑦) −1 a) Apply the given linear transformations to a line passing through the origin, given by y = k x, where k ℝ What can you say about the image obtained in each case? b) Observe the graphic results given by the tool, and answer: Which is the image of the null vector in each case? If v is a vector of the domain In each case, what is the relation between T(-v) and -T(v)? c) Can you state any conclusion? After analyzing the graphics obtained in each example, students should say that T1 is a transformation that projects each point of the plane to the line that bisects the first and third quadrant, while T2 produces a reflexion with respect to the x-axe They can also say that for every linear transformation T: V →W: 286 Marta G Caligaris, Georgina B Rodríguez and Lorena F Laugero The image of the null vector of the domain V is the null vector of the codomain W In symbols: T(0V) = 0W The image of the opossite of a vector in V is the opposite of the image of that vector In symbols: T(-v) = -T(v) Activity Given the following linear transformations: 𝑇1 : ℝ2 → ℝ2 / 𝑇1 (𝑥, 𝑦) = (𝑥 + 2𝑦, 2𝑥 + 4𝑦) 𝑥 𝑇2 : ℝ2 → ℝ2 / 𝑇2 (𝑥, 𝑦) = ( )( ) 𝑦 a) Apply the given linear transformations to a line passing through the origin, given by y = k x, where k ℝ What can you say, by watching the graph given by the tool, about the elements of the domain whose image is the null vector of the codomain? b) Check your intuition with the obtained results applying the formulae and indicate the graphical interpretation c) Are the sets obtained as solution a subspace of the domain of each transformation? When doing this activity, students will obtain the kernel of each linear transformation, without knowing about it This will be a proper situation to introduce this concept Recalling both the graphic and algebraic registers, they should complete a table as the one shown as Table Considering the obtained results, students will realize that in both cases, the kernel is a subspace of the domain of the linear transformation This situation does not only happen in these particular cases, it is a property for every linear transformation on any vector space Engaging Students in the Learning of Linear Algebra 287 Table Kernel of a linear transformation and its graphical interpretation Linear Transformation Kernel T1 ( x, y) x y;2x y Nu(T1 ) ( x; y) 2 / x y 0 Graphical Interpretation The origin of coordinates 1 T2 ( x, y ) 2 Nu(T2 ) ( x; y) 2 / y 0,5 y A line that passes through O(0, 0) 0 x 1 y CONCLUSION The use of visual tools as the ones presented during the teaching and learning processes of LA makes students compare, estimate, experiment, analize, explain, test, justify… all of them different actions of the revised Bloom´s Taxonomy Taking into account that the development of a mathematical skills system helps students to understand mathematical concepts, these visual apps are a proper tool to achieve this objective By using them, students save the time spent on doing manual calculations and they can use it for doing activities that recall mathematical skills of superior order according to Bloom´s Taxonomy, which are not usually developed in the traditional teaching of mathematics Moreover, as Williamson and Kaput say [29], an important consequence of the use of technological resources in mathematical education is that they help students to have an inductive way of thinking The students´interaction with technology produces a favourable space where students can discover mathematical relationships or properties from the observation of the repetition of certain results It also helps students to understand different concepts as they can deal with different semiotic representations in a coordinate manner, without contradictions Nevertheless the mere presence of visual tools does not ease the comprehesion process of the concepts involved in LA, reducing the formalism obstacle Professors, by their timely intervention, are the ones who will foster the meeting between students and the resource to make knowledge arise, by the design of a situation [30] Therefore, the use of visual tools as the ones designed 288 Marta G Caligaris, Georgina B Rodríguez and Lorena F Laugero should contribute to develop certain mathematical skills and construct different semiotic representations of the objects as the obstacle of formalism is mere didactic REFERENCES [1] [2] [3] [4] [5] Costa, V., & Vacchino, M (2007) La enseñanza y aprendizaje del Álgebra Lineal en la Facultad de Ingeniería, UNLP Actas del XXI Congreso Chileno de Educación en Ingeniería Universidad de Chile [Teaching and learning of Linear Algebra in the Faculty of Engineering, UNLP Proceedings of the XXI Chilean Congress of Engineering Education University of Chile] Dorier, J L., Robert, A., Robinet, J., & Rogalski, M (1997) L’Algèbre Linéaire: L’Obstacle du Formalisme Travers Diverses Recherches de 1987 1995, In J-L Dorier (Ed.), L’enseignement de l’algèbre linéaire en question (pp 105-147) Grenoble, France : La Pensée Sauvage Editions [Linear algebra: the obstacle of formalism through various researches from 1987 to 1995 In J-L Dorier (Ed.), The teaching of the linear algebra in question (pp 105-147) Grenoble, France: The Wild Thought Editions.] Uicab, R., & Oktaỗ, A (2006) Transformaciones lineales en un ambiente de geometría dinámica Revista Latinoamericana de Investigación en Matemática Educativa, (3), 459-490 [Linear transformations in an environment of dynamic geometry Latin American Journal of Research in Educational Mathematics, (3), 459-490.] Sierpinska, A., & Dreyfus, T (1999) Evaluation of a Teaching Design in Linear Algebra: The case of Linear Transformations, Recherches en Didactique des Mathématiques, 19 (1), 7-40 [Research in Mathematical Didactics, 19 (1), 7-40.] Rubio, B (2013) La enseñanza de la visualización en álgebra lineal: el caso de los espacios vectoriales cociente (tesis doctoral) Universidad Complutense de Madrid, Madrid, España [The teaching of visualization in linear algebra: the case of the quotient vector spaces (doctoral thesis) Complutense University of Madrid, Madrid, Spain.] Engaging Students in the Learning of Linear Algebra [6] 289 Caligaris, M., Rodríguez, G., Favieri, A & Laugero, L (2017) Uso de objetos de aprendizaje para el desarrollo de habilidades matemáticas Actas del XX Encuentro Nacional, XII Internacional de Educación Matemática en Carreras de Ingeniería, pp 623-631 [Use of learning objects for the development of mathematical skills Proceedings of the XX National, XII International, Meeting on Mathematics Education in Engineering Careers, pp 623-631] [7] Vinner, S (1991) The role of definitions in the teaching and learning of mathematics In D Tall (Ed.), Advanced Mathematical Thinking (pp 6581) Dordrecht: Kluwer Academic Publishers [8] Hurman, L (2007) El papel de las aplicaciones en el proceso de enseñanza – aprendizaje del Álgebra Lineal Ensanza del Álgebra Colección Digital Eudoxus Nº [The role of applications in the teaching - learning process of Linear Algebra Algebra Teaching Eudoxus Digital Collection Nº 3.] [9] Sierpinska, A (2000) On Some Aspects of Students’ Thinking in Linear Algebra In: J L Dorier (Ed.), On the Teaching of Linear Algebra (pp 209-246) Dordrecht: Springer [10] Parraguez, M y Bozt, J (2012) Conexiones entre los conceptos de dependencia e independencia lineal de vectores y el de solución de sistemas de ecuaciones lineales en R² y R³ desde el punto de vista de los modos de pensamiento Revista electrónica de investigación en educación en ciencias, 7(1), 49-72 [Connections between the concepts of linear dependence and independence of vectors and the solution of systems of linear equations in R² and R³ from the point of view of thought modes Electronic Journal of Research in Science Education, (1), 49-72.] [11] Artigue, M (1995) La enseñanza de los principios del cálculo: problemas epistemológicos, cognitivos y didácticos In P Gómez (Ed.), Ingeniería didáctica en educación matemática (pp 97-140) México: Grupo Editorial Iberoamérica [Teaching the principles of calculus: epistemological, cognitive and didactic problems In P Gómez (Ed.), Didactic Engineering in Mathematics Education (pp 97-140) Mexico: Iberoamerica Publishing Group.] 290 Marta G Caligaris, Georgina B Rodríguez and Lorena F Laugero [12] Duval, R (1998) Registros de representación semiótica y funcionamiento cognitivo del pensamiento In F Hitt (Ed.), Investigaciones en Matemática Educativa II, pp 173-201 México Cinvestav [Registers of semiotic representation and cognitive functioning of thought In F Hitt (Ed.), Research in Educational Mathematics II, p 173-201 Mexico Cinvestav] [13] Duval, R (2006) Un tema crucial en la educación matemática: La habilidad para cambiar el registro de representación, La Gaceta de la Real Sociedad Matemática Española, 9(1), 143-168 [A crucial issue in mathematics education: The ability to change the representation record, The Gazette of the Royal Spanish Mathematical Society, (1), 143 - 168.] [14] Petrovsky, A (1985) Psicología General Editorial Progreso, Moscú [General psychology Progress Editorial, Moscow.] [15] Brito Fernández, H (1987) Psicología general para los ISP La Habana, Cuba: Pueblo y Educación [General Psychology for Higher Pedagogical Institutes Havana, Cuba: People and Education.] [16] Álvarez de Zayas, C (1999) La escuela en la vida La Habana, Cuba: Pueblo y Educación [The school in life Havana, Cuba: People and Education.] [17] Bravo Estévez, M (2002) Una estrategia didáctica para la enseñanza de las demostraciones geométricas (tesis doctoral) Universidad de Oviedo, Oviedo, España [A didactic strategy for the teaching of geometric demonstrations (doctoral thesis) University of Oviedo, Oviedo, Spain.] [18] Rodríguez Rebustillo, M., & Bermúdez Sarguera, R (1993) Algunas consideraciones acerca del estudio de las habilidades Revista cubana de Psicología, 10 (1), 27-32 [Some considerations about the study of skills Cuban Journal of Psychology, 10 (1), 27-32.] [19] García Bello, B., Hernández Gallo, T., & Pérez Delgado, E (2010) The process of formation of mathematical skills Retrieved from https://es.scribd.com/document/360870457/Proceso-FormacionHabilidades-Matematicas [20] Machado Ramírez, E., & Montes de Oca Recio, N (2009) El desarrollo de habilidades investigativas en la educación superior: un acercamiento para su desarrollo Revista Humanidades Médicas, (1) [The Engaging Students in the Learning of Linear Algebra [21] [22] [23] [24] [25] [26] [27] [28] 291 development of research skills in higher education: an approach to its development Medical Humanities Magazine, (1).] Churches, A (2008) Bloom’s taxonomy for the digital age Eduteka http://eduteka.icesi.edu.co/articulos/TaxonomiaBloomDigital Bloom, B., Engelhart, M., Furst, E., Hill, W., & Krathwohl, D (1956) Taxonomy of Educational Objectives The Classification of Educational Goals Handbook Cognitive Domain New York, United States of America: Longmans Anderson, L W., & Krathwohl, D R (2001) A taxonomy for learning, teaching, and assessing, Abridged Edition Boston, MA, United States of America: Allyn and Bacon Caligaris, M., Rodríguez, G., & Laugero, L (2013) Learning Objects for Numerical Analysis Courses Procedia - Social and Behavioral Sciences, 106, 1778-1785 Caligaris, M., Rodríguez, G., & Laugero, L (2014) A Numerical Analysis Lab: Solving System of Linear Equations Procedia - Social and Behavioral Sciences, 131, 160-165 Beezer, R (2010) A first course in linear Algebra Department of Mathematics and Computer Science University of Puget Sound Washington Retrieved from http://linear.ups.edu Last visit: July 2019 Caligaris, M., Rodríguez, G., & Laugero, L (2011) Laboratorio de Álgebra lineal Autovalores y autovectores Actas del XVI Encuentro Nacional, VIII Internacional de Educación Matemática en Carreras de Ingeniería [Linear Algebra Lab Eigenvalues and eigenvectors Proceedings of the XVI National, VIII International, Meeting on Mathematics Education in Engineering Careers.] Caligaris, M., Rodríguez, G., & Laugero, L (2009) El papel de los registros semióticos en el aprendizaje de las transformaciones lineales Actas del VI Congreso Internacional de Enseñanza de la Matemática Asistida por Computadora [The role of semiotic registers in learning linear transformations Proceedings of the VI International Congress of Computer Aided Mathematics Teaching.] 292 Marta G Caligaris, Georgina B Rodríguez and Lorena F Laugero [29] Williamson, S., & Kaput, J (1999) Mathematics and virtual culture: an evolutionary perspective on technology and mathematics education Journal of Mathematical Behavior, 17(21), 265-281 [30] Sadovsky, P (2005) La Teoría de Situaciones Didácticas: un marco para pensar y actuar la enseñanza de la matemática In H Alagia, A Bressan, & P Sadovsky (Eds), Reflexiones teóricas para la educación Matemática (pp 13-65) Buenos Aires, Argentina: Libros del Zorzal [The Theory of Teaching Situations: a framework to think and act the teaching of mathematics In H Alagia, A Bressan, & P Sadovsky (Eds), Theoretical Reflections for Mathematics Education (pp 13-65) Buenos Aires, Argentina: Zorzal Books.] ABOUT THE EDITOR Ivan I Kyrchei, PhD Senior Research Fellow, Associate Professor Research Fellow of Pidstryhach Institute for Applied Problems of Mechanics and Mathematics of National Academy of Sciences of Ukraine, Ukraine Email: kyrchei@online.ua; ivankyrchei26@gmail.com Ivan Kyrchei was born in 1964 in Lviv region, Ukraine He received his PhD (Candidate of Science) degree in 2008 from Taras Shevchenko National University of Kyiv Now he is working as the Senior Researcher of Pidstryhach Institute for Applied Problems of Mechanics and Mathematics of National Academy of Sciences of Ukraine He was awarded the title of Senior Research Fellow (Algebra and The Theory of Numbers) from Ministry of Education and Science of Ukraine, equivalent 294 About the Editor to Associate Professor His research interests are mostly in Algebra, Linear Algebra and their applications His papers have published in well-known professional journals and editor's books He serves also as Editorial Board Member and reviewer in several journals INDEX A activation state variables matrix, 10, 15, 16 associated MP-matrix, 240, 245, 247 asymptotic stability, 234, 264 B behaviors, 234 C C++, 276 CGNE algorithm, 111, 112, 127 CGNR algorithm, 112, 113 chemical kinetics, 264 cognitive activity, 271 cognitive function, 290 cognitive process, 275 column determinant, 45, 49, 50, 51, 52, 54, 143, 144 column rank, 5, 12, 13 compartment analysis, 234 complete bipartite graph, 241, 246 complete graph, 241 complex eigenvalues, 261, 263 complex numbers, 259 complexity, 268, 273 computer algebra, 1, 2, 4, 21, 29, 37, 38, 40, 42 conjugate transpose, 2, 46, 50, 85, 138 cycle, 51, 241, 242, 260 D defense mechanisms, 269 determinantal rank, 54, 56 determinantal representation(s), ix, 45, 49, 50, 54, 55, 56, 57, 58, 59, 60, 61, 62, 66, 67, 68, 70, 73, 76, 79, 80, 81, 82, 83, 86, 87, 91, 93, 94, 95, 99, 106, 107, 138, 140, 143, 144, 145, 153, 158, 160 differential equations, 25, 26, 27, 37, 38, 233, 234, 235, 264 directed graph, ix, 233, 234, 235, 237, 239, 241, 243, 245, 247, 249, 250, 251, 252, 253, 255, 257, 259, 260, 261, 263, 264 directed multigraph, 236, 238 discs, 255, 258, 259, 260 Drazin inverse, 2, 3, 4, 5, 19, 32, 39, 41, 43, 44, 49, 106, 107 dynamic system, 18 dynamical systems, viii, 1, 4, 12, 21, 22, 26, 37, 38, 40, 44, 265 E eigenvalues of the matrix, 18, 278, 283 equilibrium state, 9, 10, 11 F facilitators, 267 flux balance, 252, 254, 257, 259, 261, 264 formalism obstacle, 268, 269 296 Index formation, 269, 271, 290 Frobenius norm, 6, 15, 16, 47, 112, 125, 126 full binary tree, 241, 247, 248 G gain parameter, 12 general coupled matrix equations, 111, 112, 113, 124, 130, 134 generalized bisymmetric solution, vii, 111, 112, 113, 114, 115, 117, 119, 121, 123, 124, 125, 126, 127, 129, 130, 131, 133, 135 generalized inverses, 1, 2, 3, 4, 5, 7, 9, 11, 12, 13, 15, 17, 19, 21, 23, 25, 26, 27, 29, 31, 33, 35, 37, 38, 39, 40, 41, 43, 46, 49, 137, 143, 153 Gershgorin closed disc, 256 Gradient Neural Networks (GNN), vii, 1, 2, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 23, 24, 25, 26, 27, 30, 32, 37, 43 graphs theory, 234, 235, 264 group inverse, 2, 3, 36 H Hermitian matrix, 48, 53, 54, 144 Hermitian solution, 46, 47, 48, 49, 50, 85, 86, 98, 99, 100, 102, 106, 107 HS version of BiCR algorithm, 112, 113, 114, 130 I idempotent, 61, 150 input tank, 235, 255, 256, 260 integral representation, 13, 14 internal tank, 235 involution, 48 J Java, 276 L left inverse, 3, 13 left linear combination, 52, 54 linear algebra, vii, viii, 1, 39, 40, 41, 42, 45, 103, 104, 105, 106, 109, 111, 132, 133, 137, 156, 157, 158, 159, 160, 163, 164, 167, 179, 201, 205, 209, 231, 233, 234, 235, 264, 265, 267, 268, 269, 270, 271, 273, 275, 277, 279, 281, 283, 285, 287, 288, 289, 291, 294 linear dependence, 284 linear ODE system, 237 linear systems, 264 lower matrix, 242 Lyapunov function, Lyapunov stability, M mass balance, 237, 238 Mathematica, 1, 2, 26, 27, 28, 29, 31, 32, 37, 44, 103, 109, 160, 275 mathematical skills, 268, 269, 272, 273, 274, 282, 287, 288, 289, 290 Matlab program, 1, 25, 26, 37 Matlab Simulink, 1, 25, 26, 37 matrix, vii, viii, ix, 234, 239, 240, 242, 243, 244, 245, 246, 247, 248, 249, 251, 252, 253, 254, 255, 256, 257, 258, 259, 261, 263, 277, 278, 280, 282, 283, 284, 285 matrix algebra, vii, viii matrix inverse, mixing problems, 233, 234, 264 models, vii, viii, 234 Moore-Penrose inverse, ix MP-matrix, 239, 240, 242, 243, 244, 245, 246, 247, 248, 249, 251, 252, 253, 254, 255, 256, 257, 258, 259, 261, 263 MP-matrix eigenvalues, 252, 256, 258, 261 N noncommutative determinant, 45, 50, 51 nonstandard involution, 48, 105 numerical algorithms, 3, 8, 135 Index O open system, 263 operations, 270, 272, 275, 276 ordinary differential equations, 233 orthogonal projectors, 59, 145 Outer generalized inverse, outer inverse, 2, 5, 14, 15, 19, 22, 26, 27, 28, 29, 34, 37, 39, 40, 44, 46 output tank, 235 P planar graph, 241, 244, 246 principal minor, 54, 55, 144 principal submatrix, 54 programming, 275, 276 projection matrix, 58, 68, 145 Q quaternion skew field, 46, 47, 48, 50, 106, 107, 108, 138, 159 R rank, 2, 5, 13, 14, 16, 24, 35, 36, 39, 42, 44, 46, 54, 55, 56, 61, 62, 70, 76, 86, 87, 98, 99, 138, 145, 153, 206, 207, 211, 212, 213, 214, 215, 216, 217 reflexive inverse, 46 right inverse, right linear combination, 52 row determinant, viii, 45, 49, 51, 106, 143, 160 row rank, 12, 13, 54 297 S semiotic registers, 267, 268, 269, 292 semi-scalar equivalence, ix set theory, 269 similarity of matrices, 206, 231 Simulink, 15, 16, 17, 18, 26, 29, 33 software, 275, 276 solutions qualitative behavior, 240 stability, 13, 17, 18, 43, 234, 240, 261, 262, 264 state matrix, 8, 10, 12, 18, 20, 22, 24 strictly diagonally dominant, 255 structure, 240, 241, 267 Sylvester matrix equations, viii, 47, 103, 104, 112, 113, 131, 133, 134, 139, 156, 157, 158 T tanks, 235, 237, 238 taxonomy, 274, 275, 291 technology, 287, 292 theoretical approaches, 270 transformations, 270, 276, 280, 281, 282, 284, 285, 286, 288, 292 tridimensional cube, 244 V vector, vii, 269, 270, 277, 278, 280, 281, 285, 286, 287, 289 visual apps, ix, 268, 269, 276, 287 Z Zhang Neural Networks (ZNN), 5, 7, 38, 43 ... equations Linear algebra is also used in most sciences and engineering areas, because it allows modeling many natural phenomena, and efficiently computing with such models ? ?Hot Topics in Linear Algebra? ??... 267 About the Editor 293 Index 295 P REFACE Linear algebra is the branch of mathematics concerning vector spaces and linear mappings between such spaces Systems of linear equations with several... Matrices in Chemical Problems Modeled Using Directed Graphs and Multigraphs Victor Martinez-Luaces 233 Chapter Engaging Students in the Learning of Linear Algebra Marta G Caligaris, Georgina B Rodríguez