Stefan-Maxwell and Onsager equations are equations which can be used to calculate mole flux of the component due to different types of gradients. It is assumed that driving forces of mass transfer are in equilibrium with drag forces due to interaction of different components.
While this is not math per se, I think my questions have to do much more with the math than physics or chemistry. I'll give a short introduction with some more physics background for everyone who might know more about this field.
Stefan-Maxwell equations model diffusion in gases while Onsager equations model mass transfer in electrolytes where ions can both diffuse and move in the electric field (migration). Both equations can be expressed in the same form, but I will concentrate on the Onsager type as I'm more interested in it: $$c_i \nabla \bar \mu_i = \sum_j K_{ij} (v_j - v_i) \tag {1}$$
where $\bar \mu_i$ is electrochemical potential of ion $i$, $K_{ij}$ is interaction coefficient between ions $i$ and $j$, $v$ is a velocity of the corresponding ion and $c_i$ is the concentration.
Electrochemical potential gradient takes into account both chemical and electric potential gradient and is therefore sufficient to explain mass transfer in electrolytes.
In the textbook Electrochemical Systems by Newman and Alyea, chapter 12.6: Multicomponent Transport, this equation is written in a bit of a different form on the grounds that in the system of $n$ ions, there are $n-1$ independent velocity differences $v_j - v_i$ or electrochemical potential gradients $ \nabla\bar \mu _i$.
While I understand that there are $n-1$ independent variables previously mentioned, I don't see why does that lead to the equation of this form: $$ c_i \nabla \bar \mu_i = \sum_j M_{ij}(v_j - v_0) \tag {2}$$
where $v_0$ is the velocity of any ion in the solution and $M_{ij}$ is a matrix connected to the interaction coefficient matrix $K_{ij}$ and defined as:
$$M_{ij} = \begin{cases} K_{ij}, & i \neq j \\ K_{ij} - \sum_k K_{ik}, & i=j \end{cases}$$
Equation (2) is written without a derivation or much explanation and therefore I don't understand how did we get equation (2) from equation (1).
Equation (2) is further inverted so as to express velocity differences (flux of the component) as a function of driving forces explicitly. Equation is again only written without going into much detail regarding derivation:
$$ v_j - v_0 = - \sum_{k \neq 0} L_{jk}^0 c_k \nabla \bar \mu_k \text{,}\quad j \neq 0 \tag{3} $$
where $$L^0 = -(M^0)^{-1}$$
$M^0$ is a submatrix of $M$ obtained by removing a row and a column corresponding to ion $0$ from $M$. Also, it is to be noted that $L^0$ is symmetric due to the fact that $K_{ij}$ is symmetric: $$L_{ij}^0 = L_{ji}^0 $$
I can't really figure out how was equation(3) derived from equation(2). It must have something to do with inverting matrices, but I think I miss some piece of math needed to this correctly.