principles and applications of microearthquake networks

110 5. Inversion and Optimization
then the eigenvalue problem for AT becomes
(5.19) ATV = VA
Let us take the transpose on both sides of Eq. (5.19)
(5.20) VTA = AVT
and let us postmultiply this equation by U
(5.21) VTAU = AVW
On the other hand, let us premultiply both sides of Eq. (5.16) by VT
(5.22) VTAU = VTJA
Because the left-hand sides of Eq. (5.21) and Eq. (5.22) are equal, we
obtain
(5.23) AVTU = VTUA
Let us denote the product VTU by W, whose elements are W, and write
out Eq. (5.23)
(5.24) =o
If it is assumed that the eigenvalues are distinct, we obtain W, = 0 for
i #j. This proves that VT and U are orthogonal. Since the length of
eigenvectors is arbitrary, we may choose VTU = I, where I is the identity
matrix. Thus, we have
(5.25) VT = U-I, V = (UT)-'; U = (VT-', UT = V-'
We are now in a position to derive the fundamental decomposition
theorem. If we postmultiply Eq. (5.16) by VT, we have
(5.26) AUVT = UAVT
In view of Eq. (5.25), UVT = I, so that
(5.27) A = UAVT
Equation (5.27) shows that any n x n matrix A with distinct eigenvalues is
obtainable by multiplying three matrices together, namely, the matrix U
with the eigenvectors of A as columns, the diagonal matrix A with the