Preface

Matrix theory is a branch of mathematics that has been developing over many centuries and has been successfully used in both theoretical research and applied science.

In a theoretical sense, matrix theory is a powerful tool used to develop research in such areas of mathematics as algebra, combinatorics, graph theory, statistics, and so on. It is also used to solve many engineering problems in the fields of acoustics, fluid dynamics, electromagnetics, solid mechanics, build technology, and communications.

This book consists of two sections. Section 1 contains six chapters devoted to the development of such fields of matrix theory as pencils of matrices, semi-infinite matrices, matrices with perturbed elements, the specific product of matrices, homomorphisms of matrices, and extension for informatics.

In Chapter 1, H. Huang and M.-C. Tsai study the properties of matrices products related to the k-power preserver property. The authors introduce the subject and provide a literature review. The definitive theorem of Tsai is cited to emphasize the generality of results obtained in the author's subsequent research. The generalization of theorems on the *k* -power linear preservers starts for the case of a set of general matrices on the field of complex numbers, which is proven for both the case of positive and negative powers of k. The preliminary theorem, generalizing the known theorem of Chan and Lim, is applied to prove the result. The fundamental theorem is proved for a variety of sets (spaces) of matrices, such as complex Hermitian matrices, symmetric matrices, positive definite matrices, diagonal matrices, and triangular matrices. All the considered cases differ by assumptions that apply to the properties of operators in the initial condition of the theorems.

Chapter 2, by S. Zagorodnyuk, focuses on semi-infinite matrices, generalized eigenvalue problems, and orthogonal polynomials. The classical examples are Jacobi and Hessenberg matrices, which lead to orthogonal polynomials on the real line and to orthogonal polynomials on the unit circle. Pencils of semi-infinite matrices are related to various orthogonal systems of functions. The respective polynomials are defined as generalized eigenvectors of the pencil. The polynomials under investigation have a special orthogonality relation and they are useful for a series of physical and mathematical applications. The presented examples confirm that there is a certain relation to Sobolev orthogonal polynomials that is a challenge for further investigations.

In Chapter 3, A. Kazuo proposes the optimization matrix approach for the correct reordering of sentences in linguistics. The analysis starts with the usual methods based on the transformation of data characteristics that define the correctness of the transformed sentences. The chapter describes the maximal relative sequence evaluation and applies it to resolve the problem of correcting the arrangement of the words. To do this effectively, additional means of evaluation, such as the recovery distance,

should be used. To use this approach in a PC environment, the authors use Excel. However, even though the problem data are not big in size, the number of necessary columns grows drastically. A more effective approach consists of introducing tools that decrease the required amount of memory. The illustrated application of the linearity matrix method confirms the effectiveness of the sentence reordering procedure in several examples. The evaluation of the obtained results demonstrates the possibility of using PC tools to check and correct big linguistic data.

In Chapter 4, A. N. Khimich et al. investigate the problems of weighted pseudoinverse matrices and weighted least squares (WLS). The first part of the chapter examines the sensitivity of the solution to the WLS problem with approximate initial data. The second part investigates the properties of a system of linear algebraic equations with approximate initial data and presents an algorithm for finding a weighted normal pseudosolution to the WLS problem with approximate initial data. The developed algorithm is extended for solving a WLS problem with symmetric positive semidefinite matrices and an approximate right side. The third part of the chapter analyses the exactness of the numerical solution to the WLS problem with approximate initial data, discusses the software-algorithmic approaches for improving the accuracy of computer solutions, and estimates the total error of the solution to the WLS problem.

Chapter 5, by M. Hadish is devoted to the evaluation of the errors of periodic functions by the Cesáro-Matrix product involving the conjugate Fourier series. The chapter presents an original approach related to the generalization of convergence of series that is not summarized in the classical sense. Evidence shows that the Cesáro-Matrix approach is a powerful tool for obtaining the sum of series when both the usual matrix approach and Cesáro means are not applicable. The authors prove two theorems that generalize the classical results related to slowly convergent series. Some important corollaries, which are perspective for extraction of convergence for the specific slowly convergent series, follow from the theorems. This technique has potential use in many engineering problems in which the computations lead to the calculation of slowly convergent series.

In Chapter 6, by Ivan I. Kyrchei the notions of the MPCEP inverse and CEPMP inverse are expanded to quaternion matrices and introduced new generalized inverses, the right and left MPCEPMP inverses. Direct method of their calculations, that is, their determinantal representations are obtained within the framework of theory of quaternion row-column determinants previously developed by the author. In consequence, these determinantal representations are derived in the case of complex matrices.

Section 2, consisting of six chapters, focuses on practical medicine, information theory, heat transfer, and antenna synthesis as related to the formation of COVID-19's genetic code, energy conversion processes, quantum information theory processing, solving differential and linear equations, and branching solutions to nonlinear integral equations.

In Chapter 7, S. K. Lee and M. H. Lee propose the analytical justification of the parameters of the Covid-19 genetic code predicted experimentally. This is realized by involving the information theory proof based on the doubly stochastic matrix. The genetic code model is considered in the framework of two symmetric probabilistic

channels (DNA-RNA genetic code) with different parameters of input data, which differ from the classical ones proposed by E. Chargaff. Because the computational realization of the model was not implemented until now, the authors developed a simple solution using the information theory of doubly stochastic matrix over the Shannon symmetric channel. It was proved that DNA-RNA genetic code is some kind of block circulant jacket matrix. Moreover, the chapter explores the abnormal patterns by block circulant, upper-lower, and left-right schemes that cover the distorted signal as well as the Covid-19 evolution.

In Chapter 8, G. Burel et al. demonstrate the application of linear algebra fundamentals to quantum information processing. It is shown that in many practical cases a matrix representation of the quantum systems is a powerful tool because it allows the use of linear algebra to better understand their behaviours and to better implement simulation procedures. The authors focus on Joint EigenValue Decomposition (JEVD) for the development of quantum processing. The theoretical description of the method, which aims to find a common basis of eigenvectors of a set of matrices, is supported by the effective implementation of matrix-oriented programming languages (MATLAB or Octave). It is established how to determine the encoding matrix of a quantum code from a collection of Pauli errors that opens a perspective for future study related to the interception of quantum channels and identification of the quantum coder used by a non-cooperative transmitter. Using JEVD, the existence of a subspace of the whole Hilbert space, which captures the essence of the search process, is proved. In addition, an algorithm that allows us to check this result by simulation is given.

In Chapter 9, G. F. Crosta and G. Chen model three-phase, doubly fed induction (DFI) machines by the inductance matrices with related electric and magnetic quantities. It introduces the algebraic properties of the mutual (rotor-to-stator) inductance matrix, namely, its kernel, range, and left zero divisors. An exponential representation of an inductance matrix under suitable hypotheses is derived to obtain a simple recurrent formula for the powers of the corresponding infinitesimal generator. In addition, the transformation into an exponential form is derived axiomatically. The proof of the electric torque theorem is simplified owing to newly derived formula for the product of matrices that leads in relation to the Legendre transform. As a result, a simple realistic machine model with the broken three-fold rotor symmetry is discussed and some properties for the resulting mutual inductance matrix are obtained.

A new approach to solving a non-Fourier heat equation is developed by C. N. Mihăilescu et al. in Chapter 10. This leads to the necessity to check the validity/ limits of the integral transform technique on finite domains. The proposed technique is based on the eigenvalues and eigenfunctions of the respective matrices, and its applicability to both the laser and electron beam processing problems is examined. The advantage of the method is its ability to obtain a solution with a small number of iterations and high accuracy for the like Fourier equation. However, additional efforts are needed to apply the approach proposed for the non-Fourier heat equation that is explained by the slow convergence. One such effort is applying the extra-boundary conditions. To avoid the problem with convergence, a new mixed approach is elaborated that provides the required characteristics of convergence.

In Chapter 11, M. Andriychuk focuses on the development of analytical-numerical methods for solving non-linear integral equations related to the generalized problem of phase optimization. The definitive property of such equations is that they are non-linear because of the specificity of the problem under consideration; therefore, the non-uniqueness of solutions appears. To extract a set of solutions, the respective homogeneous non-linear integral equation that results in a non-linear eigenvalue problem is used. Effective numerical algorithms are developed to find the respective eigenvalues and eigenfunctions. The study of the eigenvalues' peculiarities allows us to determine a set of points, in which the respective eigenvalues are equal to unity that determines the branching points of solutions. The total solution to the initial nonlinear equation can be presented in terms of the obtained eigenfunctions. The data of calculations testify to the ability of the proposed approach to finding solutions to non-linear integral equations numerically without large computations.

Chapter 12, by I. R. Ciric, focuses on the application of matrix differential equations for solving systems of linear algebraic equations. The accurate solutions were derived in terms of a new kind of an infinite series of matrices, which are truncated and applied repeatedly to approximate the solution. Each new term in these matrix series is obtained by multiplication on a matrix, which becomes as conditioned tending to the identity matrix that results in the effective applying the computations based on the iterative procedure. The solution method is flexible to change the initial problem's parameters. Efficient computation of an approximate solution, applicable even to poorly conditioned systems, is demonstrated based on the alternate application of two different types of minimization of associated functionals. Large computation is not needed to obtain an approximate solution for large linear systems as compared to usual methods.

It is my great pleasure to thank all book authors for their enthusiasm, patience, and improvement of the chapters throughout the reviewing process. In addition, I express my sincere thanks to Ms. Jelena Vrdoljak for her professional support during the book's preparation.

Section 1

Theory and Progress

**Mykhaylo Andriychuk,** Pidstryhach Institute for Applied Problemsof Mechanics and Mathematics, NASU, Lviv Polytechnic National University, Lviv, Ukraine
