0.2 General solutions of simultaneous equations

The second problem posed in the introduction is basically the solution of simultaneous linear equations [link] , [link] , [link] which is fundamental to linear algebra [link] , [link] , [link] and very important in diverse areas of applications in mathematics, numericalanalysis, physical and social sciences, engineering, and business. Since a system of linear equations may be over or under determined in a varietyof ways, or may be consistent but ill conditioned, a comprehensive theory turns out to be more complicated than it first appears. Indeed, there isa considerable literature on the subject of generalized inverses or pseudo-inverses . The careful statement and formulation of the general problem seems to have started with Moore [link] and Penrose [link] , [link] and developed by many others. Because the generalized solution of simultaneous equationsis often defined in terms of minimization of an equation error, the techniques are useful in a wide variety of approximation andoptimization problems [link] , [link] as well as signal processing.

The ideas are presented here in terms of finite dimensions using matrices. Many of the ideas extend to infinite dimensions using Banachand Hilbert spaces [link] , [link] , [link] in functional analysis.

The problem

Given an $M$ by $N$ real matrix $\mathbf{A}$ and an $M$ by 1 vector $\mathbf{b}$ , find the $N$ by 1 vector $\mathbf{x}$ when

or, using matrix notation,

If $\mathbf{b}$ does not lie in the range space of $\mathbf{A}$ (the space spanned by the columns of $\mathbf{A}$ ), there is no exact solution to [link] , therefore, an approximation problem can be posed by minimizing an equation error defined by

A generalized solution (or an optimal approximate solution) to [link] is usually considered to be an $\mathbf{x}$ that minimizes some norm of $\epsilon$ . If that problem does not have a unique solution, further conditions, such as also minimizing the norm of $\mathbf{x}$ , are imposed. The $l_2$ or root-mean-squared error or Euclidean norm is $\sqrt{{\epsilon }^{\mathbf{T}*}\epsilon }$ and minimization sometimes has an analytical solution. Minimization of other norms such as $l_{\infty }$ (Chebyshev) or $l_1$ require iterative solutions. The general $l_p$ norm is defined as $q$ where

for $1<p<\infty$ and a “pseudonorm" (not convex) for $0<p<1$ . These can sometimes be evaluated using IRLS (iterative reweighted least squares) algorithms [link] , [link] , [link] , [link] , [link] .

If there is a non-zero solution of the homogeneous equation

then [link] has infinitely many generalized solutions in the sense that any particular solution of [link] plus an arbitrary scalar times any non-zero solution of [link] will have the same error in [link] and, therefore, is also a generalized solution. The number of families of solutions is the dimensionof the null space of $\mathbf{A}$ .

This is analogous to the classical solution of linear, constant coefficient differential equationswhere the total solution consists of a particular solution plus arbitrary constants times the solutions to the homogeneous equation. The constants are determined from the initial(or other) conditions of the solution to the differential equation.

Ten cases to consider

Examination of the basic problem shows there are ten cases [link] listed in Figure 1 to be considered.These depend on the shape of the $M$ by $N$ real matrix $\mathbf{A}$ , the rank $r$ of $\mathbf{A}$ , and whether $\mathbf{b}$ is in the span of the columns of $\mathbf{A}$ .