Matrix Formulation of the $N$-port

If

$$\mathbf{p}^+ = \begin{bmatrix} p_1^+\\ p_2^+\\ \cdot\\ \cd... ...egin{bmatrix} p_1^-\\ p_2^-\\ \cdot\\ \cdot\\ p_N^-\\ \end{bmatrix},$$

$$\mathbf{A} = \begin{bmatrix} 1 & -1 & 0 & 0 & ... & 0 & 0\\ 0 & ... ...frac{1}{Z_4^+} & ... & \frac{1}{Z_{N-1}^+} & \frac{1}{Z_N^+}\\ \end{bmatrix},$$

$$\mathbf{B} = -\begin{bmatrix} 1 & -1 & 0 & 0 & ... & 0 & 0\\ 0 &... ...frac{1}{Z_4^-} & ... & \frac{1}{Z_{N-1}^-} & \frac{1}{Z_N^-}\\ \end{bmatrix},$$

then we can write

$$\mathbf{Ap}^+ = \mathbf{Bp}^-,$$

and

$$\mathbf{p}^- = \mathbf{B}^{-1}\mathbf{Ap}^+.$$