principles and applications of microearthquake networks

108 5. Inversion and Optimization
Geometrically, we may consider both b and x as n-dimensional vectors in
an even-determined system. Equation (5.6) says that multiplication of a
vector x by the matrix A generates a new vector b, which can be thought
of as a transformation of the original vector x. Let us investigate the case
where the new vector b happens to have the same direction as the original
vector x. In this case, b is simply proportional to x and we have the
condition
(5.7) AX = AX
Equation (5.7) is really a set of n homogeneous linear equations, i.e., (A -
AI) x = 0, where I is an identity matrix. It will have a nontrivial solution
only if the determinant of the system is zero, i.e.,
(5.8)
This determinant is a polynomial of order n in A, and thus Eq. (5.8)
leads to the characteristic equation
(5.9) A" + cn-lA"-l + c ~ - ~ A ~ ' - ~ * - + c0 = 0
In order for A to satisfy this algebraic equation, A must be one of its roots.
Since an nth order algebraic equation has exactly II roots, there are
exactly n values of A, called the eigenvalues of the matrix A, for which Eq.
(5.7) is solvable. We assume that these eigenvalues are distinct (see Wilkinson,
1965, for the general case), and write them as
(5.10) A = A,, A2, . . . , An
To every possible A = Aj, a solution of Eq. (5.7) can be found. We may
tabulate these solutions as follows
+ *
where the superscript (j) denotes the solution corresponding to Aj, and the
superscript T denotes the transpose of a vector or a matrix. These solutions
represent n distinct vectors of the n-dimensional space, and they are