In this chapter, we review some definitions and operations of matrices. Matrices play very important roles in the computation and analysis of several mathematical problems. They allow for compact notations of large sets of linear algebraic equations. Various matrix operations such as addition, multiplication, and inverses can be combined to find the required solutions in a more tractable manner. The existence of several software tools, such as MATLAB, have also made it very efficient to approach the solution by posing several problems in the form of matrix equations. Moreover, the matrices possess internal properties such as determinant, rank, trace, eigenvalues, and eigenvectors, which can help characterize the systems under consideration.

We begin with the basic notation and definitions in Section 1.1. The matrix notations introduced in this chapter are used throughout the book. Then in Section 1.2, we discuss the various matrix operations. Several matrix operations should be familiar to most readers, but some may not be as familiar, such as Kronecker products. We have classified the operations as either structural or algebraic. The structural operations are those operations that involve only the collection and arrangement of the elements. On the other hand, the algebraic operations pertain to those in which algebraic operations are implemented among the elements of a matrix or group of matrices. The properties of the different matrix operations such as associativity, com-mutativity, and distributivity properties are summarized in Section 1.3.