principles and applications of microearthquake networks

5.1. Mathematical Treatment of Linear Systems 113
sidual vector r = Ax - b equal to zero. In the least squares method we
seek a solution in which llr11* is minimized, where llrll denotes the Euclidean
length of r. In matrix notation, we write llr1p as (see Draper and
Smith, 1966, p. 58)
(5.33) llr112 = (Ax - b)T(Ax - b) = xTATAx - 2xTATb + bTb
To minimize ~ ~r~~*, partial differentiate Eq. (5.33) with respect to each component
of x and equate each result to zero. The resulting set of n equations
may be rearranged into matrix form as
(5.34) 2ATAx - 2ATb = 0
or
(5.35) ATAx = ATb
Equation (5.35) is called the system of normal equations and is an evendetermined
system. Furthermore, the matrix ATA is always symmetric,
and its eigenvalues are not only real but nonnegative. By applying the
least squares method, we not only get rid of the incompatibility of the
original equations, but also have a much nicer and smaller set of equations
to solve. For these reasons, scientists tend to use exclusively the least
squares approach for their problems. Unfortunately, there are two serious
drawbacks. In solving the normal equations by a computer, one needs
twice the computational precision of the original equations. By forming
ATA and ATb, one also destroys certain information in the original system.
We shall discuss these drawbacks later.
5.1.4.
Analysis of an Arbitrary System
Since the determinant is defined only for a square matrix, we cannot
carry out an eigenvalue analysis for a general nonsquare matrix. However,
Lanczos (1961) has shown an interesting approach to analyze an
arbitrary system. The fundamental problem to solve is
(5.36) Ax = b
where the matrix A is m X n, i.e., A has rn rows and n columns. Equation
(5.36) says that A transforms a vector x ofn components into a vector b of
rn components. Therefore matrix A is associated with two spaces: one of
rn dimensions and the other of n dimensions. Let us enlarge Eq. (5.36) by
considering also the adjoint system
(5.37) ATy = c
where the matrix AT is n X m, y is an rn-dimensional vector, and c is an