S-parameter vs impedance matrix

1.Generalized scattering parameter

\begin{align*}
a_n &=\frac{V_n ^+}{\sqrt {Z_{0n}}}\\
b_n &=\frac{V_n ^-}{\sqrt {Z_{0n}}}\\
w&here~ a_n,b_n ~are~incident~wave~and~reflected~wave,respectively\\
V_n &=V_n^+ + V_n^- =(a_n + b_n) \times \sqrt{Z_{0n}}\\
I_n &=I_n^+ + I_n^- =(V_n^+ -V_n^-)/ Z_{0n}=(a_n-b_n)/\sqrt{Z_{0n}}\\
P_n & =\frac{1}{2}Re[V_nI_n^*]=\frac{1}{2}[a^2-b^2]\\
\bf{b} & =\bf{S}\bf{a}\\
w&here~ S_{ij}=\frac{b_i}{a_j}|a_k=0~ for~ k\ne j
\end{align*}

For Z0n=Z0, ie, all ports have same characteristic impedance, we have

\begin{bmatrix} V_1 ^- \\ V_2 ^-\\.\\.\\.\\V_n ^- \end{bmatrix}=
\begin{bmatrix}S_{11}&S_{12}&... &S_{1n}\\S_{21}&S_{22}&...&S_{2n}\\.\\.\\.\\S_{n1} &S_{n2}&...&S_{nn} \end{bmatrix}
\begin{bmatrix} V_1 ^+ \\ V_2 ^+\\.\\.\\.\\V_n ^+ \end{bmatrix} \\
where \\
  S_{ij}=\frac{V_i^-}{V_j^+} |_{V_k^+ =0,~ for~ k \ne j} \\
V_k ^+=0~ --~ port~k~terminated~to~characteristic~impedance\\

For impedance matrix
[V]=[Z][I]
where

\begin{align*}
V_{ij} &=V_{ij}^+ + V_{ij}^- \\
I_{ij} &=I_{ij}^+ - I_{ij}^- \\
\end{align*}

The - sign is due to inverted direction of current \inline I^- . See below for the illustration of V+/V- and I+/I-.
I+
---->--------+-----------+
| | | |
V+ V- | port k |
| | | |
-----<-------+-----------+
I-