Mathematical Methods for Physics and Engineering : A Comprehensive Guide

8.18 SIMULTANEOUS LINEAR EQUATIONS

(a) (b)

Figure 8.1 The two possible cases whenAis of rank 2. In both cases all the

normals lie in a horizontal plane but in (a) the planes all intersect on a single

line (corresponding to an infinite number of solutions) whilst in (b) there are

no common intersection points (no solutions).

It is apparent from (8.130) that case (i) occurs when all the Cramer determinants

are zero and case (ii) occurs when at least one Cramer determinant is non-zero.

The most complicated cases with|A|= 0 are those in which the normals to the

planes themselves lie in a plane but are not parallel. In this caseAhas rank 2.

Again two possibilities exist and these are shown in figure 8.1. Just as in the

rank-1 case, if all the Cramer determinants are zero then we get an infinity of

solutions (this time on a line). Of course, in the special case in whichb= 0 (and

the system of equations is homogeneous), the planes all pass through the origin

and so they must intersect on a line through it. If at least one of the Cramer

determinants is non-zero, we get no solution.

These rules may be summarised as follows.

(i)|A|=0,b= 0 : The three planes intersect at a single point that is not the

origin, and so there is only one solution, given by both (8.122) and (8.130).

(ii)|A|=0,b= 0 : The three planes intersect at the origin only and there is

only the trivial solution,x=0.

(iii)|A|=0,b= 0 , Cramer determinants all zero: There is an infinity of

solutions either on a line ifAis rank 2, i.e. the cofactors are not all zero,

or on a plane ifAis rank 1, i.e. the cofactors are all zero.

(iv)|A|=0,b= 0 , Cramer determinants not all zero: No solutions.

(v)|A|=0,b= 0 : The three planes intersect on a line through the origin

giving an infinity of solutions.

8.18.3 Singular value decomposition

There exists a very powerful technique for dealing with a simultaneous set of

linear equationsAx=b, such as (8.118), which may be appliedwhether or not