The Method of Least Squares

Abstract

As was seen in Chapter 3, an equation of the form AX = Y, in which A and Y are matrices with the same number of rows, may have have a unique solution X, but may also have many solutions or none at all. The method of least squares is a method of choosing a “best” solution in cases where there are many, and a method of choosing a “best approximation” to a solution when there is no actual solution. As will be shown in this chapter, there is a matrix B associated with A-we will call it the mate of A-with the property that X = BY is the best choice for a solution of AX = Y (even when there is no solution) in a very specific sense. The mate of A is defined in Section 3. First, the notion of the transpose of a matrix and a few properties of transposes need to be explained.