Generalized inverses of symmetric M-matrices

In this work we carry out an exhaustive analysis of the generalized inverses of singular irreducible symmetric M-matrices. The key idea in our approach is to identify any symmetric M-matrix with a positive semi-definite Schrödinger operator on a connected network whose conductances are given by the off-diagonal elements of the M-matrix. Moreover, the potential of the operator is determined by the positive eigenvector of the M-matrix. We prove that any generalized inverse can be obtained throughout a Green kernel plus some projection operators related to the positive eigenfunction. Moreover, we use the discrete Potential Theory associated with any positive semi-definite Schrödinger operator to get an explicit expression for any generalized inverse, in terms of equilibrium measures. Finally, we particularize the obtained result to the cases of tridiagonal matrices and circulant matrices.