Preface

Matrices have applications in a huge number of scientific fields. In physics, they are used to study physical phenomena, such as the motion of rigid bodies. In computer graphics, they are used to manipulate 3D models and project them onto a 2D screen. In probability theory and statistics, stochastic matrices are used to describe sets of probabilities. Matrices calculus may be used in economics to describe systems of economic relationships. A main part of numerical analysis focuses on the development of efficient algorithms for matrix computations, a subject that is centuries old and is today an expanding area of research. Matrix decomposition meth‐ ods simplify computations, both theoretically and practically. Algorithms that are tailored to particular matrix structures, such as sparse matrices and near-diagonal matrices, expedite computations in the finite element method and other computations. Infinite matrices are present in planetary theory and atomic theory. They are the matrices representing the deriva‐ tive operators, which act on the Taylor series of a function. This new book reviews current research, including applications of matrices and spaces, as well as other characteristics.

The book is divided into two sections. The first section (Chapters 1 and 2) discusses the ap‐ plication of matrices that has become an area of academic research and of great importance in many scientific fields. In Chapter 1, within the framework of the theory of row/column determinants, the determinantal representations (analogs of Cramer's rule) of a partial solu‐ tion to the system of two-sided quaternion matrix equations, *A*1XB1=*C*1, *A*2XB2=*C*<sup>2</sup> are ana‐ lyzed. It also gives Cramer's rules for its special cases with one-sided equations and considers the two systems with the first equation *A*1*X*=*C*1 and XB1=*C*1, respectively, and with an unchanging second equation. Cramer's rules for special cases when two equations are one sided, to wit, the system of equations *A*1*X*=*C*1, XB2=*C*<sup>2</sup> and the system of the equations *A*1*X*=*C*1, *A*2*X*=*C*2, are studied as well. Chapter 2 introduces and studies a matrix that has the exponential function as one of its eigenvectors and realizes that this matrix represents finite difference derivation of vectors on a partition. This matrix leads to new expressions for finite difference derivatives, which are exact for the exponential function. A number of properties of this matrix, induced derivatives, and its inverse are also found. In addition, the expres‐ sion for the derivative of a product, a ratio, and the inverse of vectors plus the equivalent of the summation by parts theorem of continuous functions are also described. This matrix could be of interest to discrete quantum mechanics theory.

The second section (Chapters 3 to 5) comprises three chapters discussing spaces and linear systems. In Chapter 3, mixing problems are considered since they always lead to linear ODE systems, and the corresponding associated matrices have different structures that deserve to be studied in depth. This structure depends on whether there is recirculation of fluids and if the system is open or closed, among other characteristics such as the number of tanks and their internal connections. Several statements regarding matrix eigenvalues are analyzed for different structures and also some questions and conjectures are posed. Finally, qualitative remarks about differential equation system solutions and their stability or asymptotical sta‐ bility are included. In Chapter 4, special compound ´ magic squares are considered and a dimensional subspace of the nullspace of the ´ squares is determined. All vectors in the subspaces possess the property that the sum of all entries of each vector equals zero. In Chapter 5, a new type of regular matrix generated by Fibonacci numbers is introduced and we shall investigate its various topological properties. The concept of mathematical regulari‐ ty in terms of Fibonacci numbers and phyllotaxy have been discussed.

> **Prof. Dr. Hassan A. Yasser** Thi-Qar University Science College Physics Department Iraq

**Section 1**

**Applications of Matrices**
