I have a fat Toeplitz matrix, say,
\begin{equation*} T = \begin{pmatrix} 1 & 1 & 1 & 1 & 1 & 0 & 0 & 0 & 0 \\ 0 & 1 & 1 & 1 & 1 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 1 & 1 & 1 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 1 & 1 & 1 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 & 1 & 1 & 1 & 1 \end{pmatrix} \end{equation*}
and I would like to find the pseudo-inverse, $T^+$, to solve the linear system $Tx = y$ for $x$.
Is finding $T^+$ for $\hat{x} = T^+ y$ as simple finding that which corresponds to the minimum norm solution, $T^T(TT^T)^{-1}$, or its regularized version, $T^T(TT^T+\lambda)^{-1}$, where $\lambda$ is a regularization parameter? Is there anything special about the Toeplitz structure that can be taken into account to find the inverse?
