I've got the following system of equations: $$ Mx = B^T d$$ where $$M = B^T C B \in \mathbb R^{m \times m}$$, $$B \in \mathbb R^{n \times m}$$ an incidence matrix of a closed, connected network, $$ C \in \mathbb R^{m \times m}$$ the indicator matrix and $$ d \in \mathbb R^n, x \in \mathbb R^m$$.
I know and already prooved that M is symmetric and positive semidefinite and that C is symmetric positive definite. In addition, I know that the system of equations doesn't have a unique solution.
Now my question: How does the system of equation looks like if I erase the i-th row and the i-th column of M?
Thanks for any hint!