I want to apply Gauss-Seidel method in a least square deconvolution problem. The convolution of two vectors is written in: $h * x = z$.
$$z(n) = \sum_{i=0}^{N-1}h(i)x(n-i)$$
It is a linear transform and can be represented as a matrix multiplies $x$:$H x = z$,where $H$ is constructed from $h$.
We want to extract estimated solution of $x$ with known $h$ and $z$, which makes up a least square problem: $x = (H^TH)^{-1}H^T z$. The solution can be obtained by solving the linear system
$$Ux = H^T z \quad and \quad U = H^TH$$
I wonder if the Gauss-Seidel procedure convergence in this case, which seems always to be in experiment. The matrix $U$ is in very special form. It is symmetric and diagonal-constant; the elements are in great relationship with the self-correlation of $h$. I doubt if we can prove the positive definite of $U$ to prove the convergence of Gauss-Seidel procedure.