I have a system that $\mathbf{A}\mathbf{x}+\mathbf{n}=\mathbf{b}$.
$\mathbf{A}^{mn}\in\mathbb{R}$ is toeplitz matrix, and $\mathbf{x}$ and $\mathbf{n}$ are unknown. I am looking for a decomposition for $\mathbf{A}$ so that I can write $\mathbf{A}=\mathbf{B}\mathbf{C}$ where $\mathbf{B},\mathbf{C}$ are have a special structure that helps in finding $\mathbf{x}$
$$ \mathbf{A} = \begin{bmatrix} a & b & c & 0 \ldots & \ldots & 0 \\\ 0 & a & b & \ddots & & \vdots \\\ 0 & 0 & a & \ddots & \ddots& \vdots \\\ \vdots & \ddots & \ddots & a & b & c\end{bmatrix} $$
I searched over net and found work related to this but I couldn't figure out where should I start from. Any lead will be much. Reference to any good book that specifically addresses this problem will be much appreciated
Your matrix has a special structure, the degree of freedom of the matrix is just 3 no matter how big $n$ is ( assuming $A_{n\times n}$). Just three numbers , $a,b,$ and $c$ define the entire matrix, so it is likely that you may find a closed form solution that involves those variables, and indeed there is.
Since you have a tridiagonal matrix that is also a toeplitz, there is a simple closed form solution, the eigenvalues of a Tridiagonal Toeplitz is given by, $$\lambda_k = a + \sqrt{bc}\cos{\frac{k\pi}{n+1}}, k = 1,2,\cdots n$$
The eigenvalues are allocated symmetrically with respect to $a$.
Once you know eigenvalues you can do an eigenvalue decomposition, which is a direct approach rather using iterative solver. Such matrices are also called banded Toeplitz matrices (see. Banded symmetric Toeplitz).