Mathematical Methods for Physics and Engineering : A Comprehensive Guide

(lu) #1

MATRICES AND VECTOR SPACES


Show that the set of simultaneous equations
2 x 1 +4x 2 +3x 3 =4,
x 1 − 2 x 2 − 2 x 3 =0, (8.123)
− 3 x 1 +3x 2 +2x 3 =− 7 ,
has a unique solution, and find that solution.

The simultaneous equations can be represented by the matrix equationAx=b,i.e.



243


1 − 2 − 2


−33 2






x 1
x 2
x 3


=




4


0


− 7



.


As we have already shown thatA−^1 exists and have calculated it, see (8.59), it follows that
x=A−^1 bor, more explicitly, that



x 1
x 2
x 3


=^1


11




21 − 2


4137


− 3 − 18 − 8






4


0


− 7



=




2


− 3


4



. (8.124)


Thus the unique solution isx 1 =2,x 2 =−3,x 3 =4.


LU decomposition

Although conceptually simple, finding the solution by calculatingA−^1 can be


computationally demanding, especially whenNis large. In fact, as we shall now


show, it is not necessary to perform the full inversion ofAin order to solve the


simultaneous equationsAx=b. Rather, we can perform adecompositionof the


matrix into the product of a squarelower triangularmatrixLand a squareupper


triangularmatrixU, which are such that


A=LU, (8.125)

and then use the fact that triangular systems of equations can be solved very


simply.


We must begin, therefore, by finding the matricesLandUsuch that (8.125)

is satisfied. This may be achieved straightforwardly by writing out (8.125) in


component form. For illustration, let us consider the 3×3 case. It is, in fact,


always possible, and convenient, to take the diagonal elements ofLas unity, so


we have


A=



100
L 21 10
L 31 L 32 1





U 11 U 12 U 13
0 U 22 U 23
00 U 33



=



U 11 U 12 U 13
L 21 U 11 L 21 U 12 +U 22 L 21 U 13 +U 23
L 31 U 11 L 31 U 12 +L 32 U 22 L 31 U 13 +L 32 U 23 +U 33


 (8.126)

The nine unknown elements ofLandUcan now be determined by equating

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