Volume of a Complex Matrix and Its Applications in Oriented Graphs

For a complex matrix A of rank $r>0$, the volume of A is the product of r nonzero singular values of A. We prove that the volume of A is the square root of the sum of squares of modulus of all r× r minors of A. Ben-Israel provided a formula for the Moore-Penrose inverse of A when A is a real matrix using the volumes of A and some of its submatrices. We generalize the result to complex matrices. As applications, we find the volume of a complex incidence matrix of an oriented graph with complex edge weights for oriented trees and unicyclic graphs. We also present a combinatorial formula for the complex Laplacian matrix of a weakly connected oriented graph with complex edge weights.