Mathematical Methods for Physics and Engineering : A Comprehensive Guide

(Darren Dugan) #1

8.18 SIMULTANEOUS LINEAR EQUATIONS


anyvector in the null space ofA(i.e.Ay= 0 )then


A(x+y)=Ax+Ay=Ax+ 0 =b,

and sox+yis also a solution. Since the null space is (n−r)-dimensional, so too


is the space of solutions.


We may use the above results to investigate the special case of the solution of

ahomogeneousset of linear equations, for whichb= 0. Clearly the setalwayshas


the trivial solutionx 1 =x 2 =···=xn=0,andifr=Nthis will be the only


solution. Ifr<N, however, there are infinitely many solutions; they form the


null space ofA, which has dimensionn−r. In particular, we note that ifM<N


(i.e. there are fewer equations than unknowns) thenr<Nautomatically. Hence a


set ofhomogeneouslinear equations with fewer equations than unknownsalways


has infinitely many solutions.


8.18.2Nsimultaneous linear equations inNunknowns

A special case of (8.118) occurs whenM=N. In this case the matrixAissquare


and we have the same number of equations as unknowns. SinceAis square, the


conditionr=Ncorresponds to|A|= 0 and the matrixAisnon-singular.The


caser<Ncorresponds to|A|=0,inwhichcaseAissingular.


As mentioned above, the equations will have a solution providedblies in the

range ofA. If this is true then the equations will possess a unique solution when


|A|= 0 or infinitely many solutions when|A|= 0. There exist several methods


for obtaining the solution(s). Perhaps the most elementary method isGaussian


elimination; this method is discussed in subsection 27.3.1, where we also address


numerical subtleties such as equation interchange (pivoting). In this subsection,


we will outline three further methods for solving a square set of simultaneous


linear equations.


Direct inversion

SinceAis square it will possess an inverse, provided|A|= 0. Thus, ifAis


non-singular, we immediately obtain


x=A−^1 b (8.122)

as the unique solution to the set of equations. However, ifb= 0 then we see


immediately that the set of equations possesses only the trivial solutionx= 0 .The


direct inversion method has the advantage that, onceA−^1 has been calculated,


one may obtain the solutionsxcorresponding to different vectorsb 1 ,b 2 , ...on


the RHS, with little further work.

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