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 $r\times r$ 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.