Mathematical Methods for Physics and Engineering - Matematica.NET

8.18 SIMULTANEOUS LINEAR EQUATIONS(a)(b)Figure 8.1 The two possible cases when A is of rank 2. In both cases all thenormals lie in a horizontal plane but in (a) the planes all intersect on a singleline (corresponding to an infinite number of solutions) whilst in (b) there areno common intersection points (no solutions).It is apparent from (8.130) that case (i) occurs when all the Cramer determinantsare 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 theplanes themselves lie in a plane but are not parallel. In this case A has rank 2.Again two possibilities exist and these are shown in figure 8.1. Just as in therank-1 case, if all the Cramer determinants are zero then we get an infinity ofsolutions (this time on a line). Of course, in the special case in which b = 0 (andthe system of equations is homogeneous), the planes all pass through the originand so they must intersect on a line through it. If at least one of the Cramerdeterminants 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 theorigin, 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 isonly the trivial solution, x =0.(iii) |A| =0,b ≠ 0, Cramer determinants all zero: There is an infinity ofsolutions either on a line if A is rank 2, i.e. the cofactors are not all zero,or on a plane if A is 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 origingiving an infinity of solutions.8.18.3 Singular value decompositionThere exists a very powerful technique for dealing with a simultaneous set oflinear equations Ax = b, such as (8.118), which may be applied whether or not301