A Symmetric Rank-Revealing Toeplitz Matrix Decomposition

. In signal and image processing, regularization often requires a rank-revealing decomposition of a symmetric Toeplitz matrix with a small rank deficiency. In this paper, we present an efficient factorization method that exploits symmetry as well as the rank and Toeplitz properties of the given matrix. Keywords: Toeplitz matrix, regularization, symmetric rank-revealing decomposition 1. Introduction In signal and image processing applications [5], [6], a noisy and distorted signal vector x is given by x = Tx + w; (1) where x and w represent an unknown original signal vector and a noise vector, respectively, and T is a predetermined matrix describing the spread of signals. This problem arises often in array processing, where the matrix T may be real, symmetric, and Toeplitz. Assuming the dimensions of T to be n \Theta n, we have T = 0 B B B B @ t 1 t 2 t 3 : : : t n t 2 t 1 t 2 : : : t n\Gamma1 t 3 t 2 t 1 : : : t n\Gamma2 . . . . . . . . . . . . . . . t n t n\Gamma1 t n\Gamm..