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Linear Systems ofEquations 9
Although the Gauss-Jordan method considered here is well known, it is less
well known that the real-coefficient case can easily be extended to solve
systems having both complex coefficients and variables. BASIC language
Program B2-1 for the Gauss-Jordan method is contained in Appendix B, and
its preamble for coping with the complex system case is Program B2-2.
2.2.1. The Gauss-Jordan Elimination Method. The Gauss-Jordan elimination
method is but one of several acceptable means to solve systems of real
linear equations (see Hamming, 1973, for a commentary). The problem to be
solved is to find x when
in matrix notation, or, written out,
Ax=a,
a11x 1+ al2x2+ aJ3x J = a14 ,
a2,x, + a22x 2 -+. a2Jx J
= a 24 ,
aJJx, + aJ2x2+ a J 3x 3 = a34 .
(2.3)
(2.4)
The order N = 3 case will be discussed without loss of generality. Readers not
familiar with the matrix notation in (2.3) are urged to refer to an introductory
book on linear algebra, such as that of Noble (1969). There will be frequent
need for this shorthand notation, although the subject will not be much more
rigorous than understanding the equivalence of (2.3) and (2.4). It is also
helpful to sketch the N = 2 case of two lines in the XI - x 2
plane and to recall
that the solution is merely the intersection of these two lines. The concept
extends to hyperplanes in N-dimensional space.
The Gauss-Jordan algorithm evolves (2.4) into the solution form
x, +0+0=bI4,
0+x,+0=b'4'
(2.5)
0+0+x,=b'4'
by scaling adjacent rows so that'subtractions between rows produce the zeros
in the columns in (2.5), working from left to right. Recall that scaling a given
equation or adding one to another does not change a system of linear
equations.
A specific example (Ley, 1970) begins with the "augmented" matrix formed
from (2.4):
Consider the array
a"
[all
a 13
a 21 a 22 a 23 a 24 .
a 31 an a" a 34
[-!
-I
a 14 ] .
I
- I ]
I I 6 .
I -I 2
(2.6)
(2.7)