Matrix theory is a powerful field of mathematics that has found applications in the solution of several real-world problems, ranging from the solution of algebraic equations to the solution of differential equations. Its importance has also been enhanced by the rapid development of several computer programs that have improved the efficiency of matrix analysis and the solution of matrix equations.
We have allotted three chapters to discussing matrix theory. Chapter 1 contains the basic notations and operations. These include conventions and notations for the various structural, algebraic, differential, and integral operations. As such, this chapter focuses on how to formulate problems in terms of matrix equations, the various approaches of matrix algebraic manipulations, and matrix partitions.
Chapter 2 then focuses on the solution of the linear equation given by Ax = b, and it includes both direct and indirect methods. The most direct method is to find the inverse of A and then evaluate x = A−1 b. However, the major practical issue is that matrix inverses become unwieldy when the matrices are large. This chapter is concerned with finding the solutions by reformulating the problem to take advantage of available matrix properties. Direct methods use various factorizations of A based on matrices that are more easily invertible, whereas indirect methods use an iterative process starting with an initial guess of the solution. The methods can then be applied to linear least-squares problems, as well as to the solution of multivariable nonlinear equations.