Below you can find the problem that I am trying to understand. My main problem is to understand where do the reparametrization that they propose come from. Alternatively I have try to connect it with the constraint LS KKT solution but I do not see the connection.
$$ \begin{aligned} & \mathbf{r}_t=\left(\begin{array}{l:l} \mathbf{1} & \mathbf{B}_i \end{array}\right) \mathbf{f}_{m i, t}+\boldsymbol{\varepsilon}_t, \quad t=1,2, \cdots, T \\ & =\mathbf{B}_{m i} \mathbf{f}_{m i, t}+\boldsymbol{\varepsilon}_t \\ & \end{aligned} $$
where $\mathbf{B}_{m i}$ is an $N \times(K+1)$ matrix with $\mathbf{B}_i$ an $N \times K$ matrix of $1^{\prime} s$ and $0^{\prime} s$ representing $K$ industry sectors, with each stock belonging to one and only one sector over the given time interval, and $$ \mathbf{f}_{m i, t}=\left(f_{0, t}, f_{1, t},, f_{2, t}, \cdots, f_{K, t}\right)^{\prime} . $$
As such the matrix B is rank deficient with rank K instead of K+1. Consequently the use of LS to fit the above model for each time period does not lead to a unique solution. One solution is to impose a constraint:
$$ f_{1, t}+f_{2, t}+\cdots+f_{K, t}=0 . $$
This can be accomplished with the reparameterization (this is where I don't understand) $$ \mathbf{f}_{m i, t}=\mathbf{R}_{m i} \mathbf{g}_{m i, t} $$ $$ \mathbf{r}_t=\mathbf{B}_{m i} \mathbf{R}_{m i} \mathbf{g}_{m i, t}+\boldsymbol{\varepsilon}_t $$
where $$ \mathbf{R}_{m i}=\left(\begin{array}{c} \mathbf{I}_K \\ \boldsymbol{a}^{\prime} \end{array}\right) \sim(K+1) \times K $$ $$ \boldsymbol{a}^{\prime}=(0,-1,-1, \cdots,-1) \sim K \times 1 $$ $$ \mathbf{g}_{m i, t}=\left(g_{1, t}, g_{2, t}, \cdots, g_{K, t}\right)^{\prime} . $$
The model (9) now has a unique least-squares solution $\hat{\mathbf{g}}_t$, and it is easy to check that (8) insures that the constraint (7) is satisfied.
