Some matrix factorization identities for discrete inverse scattering

AbstractThis note analyzes the relationships between several inverse-scattering methods and points out that all classical approaches implicitly construct a (lower-upper) triangular factorization of a given positive definite Toeplitz matrix. It is shown that the various inversion methods implicitly obtain the factorization by solving different nested sets of linear equations and expressing the factors in terms of the solutions obtained. The fact that the triangular factorization of a matrix with nonzero leading minors is unique immediately yields various identities, e.g., involving the so-called “central mass” sequence, that were derived in the literature with considerably more manipulation. In fact, our basic factorization results are somewhat more general, since they do not require the Toeplitz assumption