Mathematical Methods for Physics and Engineering : A Comprehensive Guide

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.