principles and applications of microearthquake networks

114 5. Inversion and Optimization
n-dimensional vector. We now combine Eq. (5.36) and Eq. (5.37) as follows:
(5.38)
Now, this combined system has a symmetric matrix, and one can proceed
to perform an eigenvalue analysis like that described before (for details,
see Lanczos, 1961, pp. 115-123). Finally, one arrives at a decomposition
theorem similar to Eq. (5.27) for a real m X n matrix A with rn 2 n
(5.39) A = USVT
where
(5.40) UTU = I,, V T = I,
and
(5.41)
The rn X rn matrix U consists of rn orthonormalized eigenvectors of AAT,
and the n X n matrix V consists of n orthonormalized eigenvectors of ATA.
Matrices I, and 1, are rn x rn and n x n identity matrices, respectively.
The matrix S is an rn X n diagonal matrix with off-diagonal elements Sij =
0 for i # j , and diagonal elements Sii = ui, where ui, i = 1, 2, . . . , n are
the nonnegative square roots of the eigenvalues of ATA. These diagonal
elements are called singular values and are arranged in Eq. (5.41) such
that
(5.42) (T12.(T22 ... ?(T,LO
The above decomposition is known as singular value decomposition
(SVD). It was proved by J. J. Sylvester in 1889 for square real matrices
and by Eckart and Young (1939) for general matrices. Most modern texts
on matrices (e.g., Ben-Israel and Greville, 1974, pp. 242-251; Forsythe
and Moler, 1967, pp. 5-11; G. W. Stewart, 1973, pp. 317-326) give a
derivation of the singular value decomposition. The form we give here
follows that given by Forsythe et crl. (1977, p. 203). Lanczos (1961, pp.
120-123) did not introduce the singular values explicitly, but used the term