In Chapter 1, we started with the study of matrices based on their composition, structure, and basic mathematical operations such as addition, multiplication, inverses, determinants, and so forth, including matrix calculus operations. Then, in Chapter 2, we focused on the use of matrices to solve simultaneous equations of the form Ax = b, including their applications toward the solution of nonlinear equations via Newton algorithms. Based on linear algebra, we saw that Ax can also be taken to be a linear combination of the columns A with the elements of x acting as the weights. Under this perspective, the least-squares problem shifted the objective to be that of finding x that would minimize the residual error given by r = b − Ax.

In this chapter, we return to the equation Ax = b with a third perspective. Here, we consider matrix A to be an operator that will “transform” (or “map”) an input vector x to yield an output vector b, as shown schematically in Figure 3.1. We call this the matrix operator perspective of the linear equation. The main focus is now on A as a “machine” that needs to be analyzed, constructed, or modified to achieve some desired operational characteristics. For instance, we may want to construct a matrix A that rotates, stretches, or flips various points xi described by vectors. As another example, a stress tensor (to be discussed in Chapter 4) can be represented by a matrix T, which can then be used to find the stress vector s pointing in the direction of a unit vector n by the operation s= Tn.