principles and applications of microearthquake networks

5.1. Mathematical Treatment of Linear Systems 109
called the eigenvectors of the matrix A. We may denote them by ul, u2,
. . . , un, where
u1 = (xy, xi'), . . . , xn (1)) T
(5.12)
u2 = (xi", xy', . . . , X 'y
u, = (x:"), x?'. . . . , xy)T
Because Eq. (5.7) is valid for each A = A,, J = 1, . . . , n, we have
(5.13)
Auj = Ajuj,
j = 1, . . . , n
In order to write these n equations in matrix notation, we introduce the
following definitions:
(5.14) A=
(5.15)
In other words, A is a diagonal matrix with the eigenvalues of matrix A as
diagonal elements, and U is an n X n matrix with the eigenvectors of
matrix A as columns. Equation (5.13) now becomes
(5.16) AU = UA
Let us carry out a similar analysis for the transposed matrix AT whose
elements are defined by
(5.17) ATj = A ji
Because any square matrix and its transpose have the same determinant,
both matrix A and its transpose AT satisfy the same characteristic equation
(5.9). Consequently, the eigenvalues of AT are identical with the
eigenvalues of A. If we let vl, v2, . . . , Vn denote the eigenvectors of AT
and define
' (5.18) v= i' v1v2 ...