Given some measurements on empirical data (in the form of a multigraph with two weighted edges between every pair of vertices), I would like to place the measurements in a Hermitian matrix that also (1) resembles a Laplacian matrix and (2) is positive semi-definite.
The goal is to measure algebraic connectivity (i.e., the second smallest eigenvalue) while simultaneously considering two types of edge weights.
My measurements are both on the [0,1] interval and I think this is a problem. From what I have read (Wilson and Hancock, Spectral Analysis of Complex Laplacian Matrices), each entry on the diagonal must be greater than the sum of magnitudes of the off-diagonal entries on the same row/column. They seem to recommend adding some positive constant, if necessary.
Can anyone recommend a procedure for obtaining this matrix? In particular, how should I scale my measurements? Should my measurements be treated as the magnitude and phase of the complex entries (as recommended by the reference above) or as the real and imaginary parts (my original hunch)?
If I understand you correctly, one of the matrices you want will be positive semidefinite.
In particular, let $w^{(1)}_{jk} = w^{(1)}_{kj}$ and $w^{(2)}_{jk}=w^{(2)}_{kj}$ be the weights between vertices $j$ and $k$. We can define the matrix $M$ by $$ M_{jk} = -w^{(1)}_{jk} - i \, w^{(2)}_{jk} \quad j<k\\ M_{kk} = \sum_{j=1}^n \sqrt{(w^{(1)}_{jk})^2 + (w^{(2)}_{jk})^2}\\ M_{jk} = \overline{M_{kj}} $$ This matrix is directly analogous to the Laplacian.