& {\widetilde}{\Lambda}_{c,n-1,n-1}^{-} & {\widetilde}{\Lambda}_{c,n-1,n}^{-} \\
    (n,\mathbf{E}^{+}) & 0 & 0 & \cdots & 0 & {\widetilde}{\Lambda}_{c,nn}^{-}
  },\end{aligned}$$ $$\begin{aligned}
\label{eq:blockG+}
  {\widetilde}{\mG}_{c}^{+}=
  \kbordermatrix{
    & (0,\mathbf{E}^{+}) & (1,\mathbf{E}^{+}) & \cdots & (n-1,\mathbf{E}^{+}) & (n,\mathbf{E}^{+}) \\
    (0,\mathbf{E}^{+}) & {\widetilde}{\mG}_{c,00}^{+} & {\widetilde}{\mG}_{c,01}^{+} & \cdots & {\widetilde}{\mG}_{c,0,n-1}^{+} & {\widetilde}{\mG}_{c,0n}^{+} \\
    (1,\mathbf{E}^{+}) & 0 & {\widetilde}{\mG}_{c,11}^{+} & \cdots & {\widetilde}{\mG}_{c,1,n-1}^{+} & {\widetilde}{\mG}_{c,1n}^{+} \\
    \vdots & \vdots & \vdots & \ddots & \vdots & \vdots \\
    (n-1,\mathbf{E}^{+}) & 0 & 0 & \cdots & {\widetilde}{\mG}_{c,n-1,n-1}^{+} & {\widetilde}{\mG}_{c,n-1,n}^{+} \\
    (n,\mathbf{E}^{+}) & 0 & 0 & \cdots & 0 & {\widetilde}{\mG}_{c,nn}^{+}
  },\end{aligned}$$ and $$\begin{aligned}
\label{eq:blockG-}
  {\widetilde}{\mG}_{c}^{-}=
  \kbordermatrix{
    & (0,\mathbf{E}^{-}) & (1,\mathbf{E}^{-}) & \cdots & (n-1,\mathbf{E}^{-}) & (n,\mathbf{E}^{-}) \\
    (0,\mathbf{E}^{-}) & {\widetilde}{\mG}_{c,00}^{-} & {\widetilde}{\mG}_{c,01}^{-} & \cdots & {\widetilde}{\mG}_{c,0,n-1}^{-} & {\widetilde}{\mG}_{c,0n}^{-} \\
    (1,\mathbf{E}^{-}) & 0 & {\widetilde}{\mG}_{c,11}^{-} & \cdots & {\widetilde}{\mG}_{c,1,n-1}^{-} & {\widetilde}{\mG}_{c,1n}^{-} \\
    \vdots & \vdots & \vdots & \ddots & \vdots & \vdots \\
    (n-1,\mathbf{E}^{-}) & 0 & 0 & \cdots & {\widetilde}{\mG}_{c,n-1,n-1}^{-} & {\widetilde}{\mG}_{c,n-1,n}^{-} \\
    (n,\mathbf{E}^{-}) & 0 & 0 & \cdots &0 & {\widetilde}{\mG}_{c,nn}^{-}
  }.\end{aligned}$$ Plugging – into and then comparing all the block entries on both sides, we end up with the following procedure to compute the factorization recursively.

In accordance to Theorem \[thm:WHZ\], for any generator 