principles and applications of microearthquake networks

5.1. Mathematical Treatment of Linear Systems 107
the overdetermined set of equations into an even-determined one by the
least-squares method. Thus on the surface, one may think that n equations
are sufficient to determine n unknowns. Unfortunately, this is not always
true, as illustrated by the following example:
(5.3)
x1 + x p + x3 = I, XI - x, + x3 = 3
2x, + 2x, + 2x3 = 2
These three equations are not sufficient to determine the three unknowns
because there are actually only two equations, the third equation being a
mere repetition of the first. Readers may notice that the determinant for
the system of equations in Eq. (5.3) is zero, and hence the system is
singular and no inverse exists. However, the solvability of a linear system
depends on more than the value of the determinant. For example, a small
value for a determinant does not mean that the system is difficult to solve.
Let us consider the following system, in which the off-diagonal elements
of the matrix are zero:
The determinant is 10-loo, which is very small indeed. But the solution is
simple, i.e., xi = xz = . - .
-
xloo = 10. On the other hand, a reasonable
determinant does not mean that the solution is stable. For example,
(5.5)
leads to a simple solution of x = (A). and the determinant is 1. However, if
we perturb the right-hand side to b = ($:A), the solution becomes x = (-4:;).
In other words, if h, is changed by 555, the solution is entirely different.
5.1.1.
Analysis of an Even-Determined System
The preceding discussion clearly demonstrates that solving n linear
equations for n unknowns is not straightforward, because we need to
know some critical properties of the matrix A. We now present an analysis
of the properties of A, and our treatment here closely follows Lanczos
(1956, pp. 52-79). We first write Eq. (5.1) in reversed sequence as
(5.6) b = Ax