# 0.2 General solutions of simultaneous equations

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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

$\left[\begin{array}{ccccc}{a}_{11}& {a}_{12}& {a}_{13}& \cdots & {a}_{1N}\\ {a}_{21}& {a}_{22}& {a}_{23}& & \\ {a}_{31}& {a}_{32}& {a}_{33}& & \\ ⋮& & & & ⋮\\ {a}_{M1}& & & \cdots & {a}_{MN}\end{array}\right]\left[\begin{array}{c}{x}_{1}\\ {x}_{2}\\ {x}_{3}\\ ⋮\\ {x}_{N}\end{array}\right]=\left[\begin{array}{c}{b}_{1}\\ {b}_{2}\\ {b}_{3}\\ ⋮\\ {b}_{M}\end{array}\right]$

or, using matrix notation,

$\mathbf{A}\mathbf{x}=\mathbf{b}$

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

$\epsilon =\mathbf{A}\mathbf{x}-\mathbf{b}.$

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

$q={||x||}_{p}=\left(\sum _{n}{|x\left(n\right)|}^{p}{\right)}^{1/p}$

for $1 and a “pseudonorm" (not convex) for $0 . 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

$\mathbf{A}\mathbf{x}=\mathbf{0},$

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 $\mathbf{A}$ .

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