Definition. Kernel complement and pseudodeterminant [fdip-0031]

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Let \(L:E\to E\) be a self-adjoint positive semidefinite operator on a finite real inner-product space \((E,h)\). Set \[\begin {aligned} K&=\ker L,& H&=K^\perp ,\\ k&=\dim K,& r&=\dim H. \end {aligned}\] Write \(i:H\to E\) for inclusion, \(p_H:E\to H\) for orthogonal projection, \(P_H=ip_H\) and \(P_K=I-P_H\) for the endomorphism projections.

The spectral theorem gives \(E=K\mathbin \perp H\). The positive-eigenvalue vectors span \(H\), so \(L(H)\subset H\) and \(L_H=L|_H:H\to H\) is positive definite. Define the pseudodeterminant by \[\operatorname {pdet}L=\det L_H =\prod _{\lambda _a>0}\lambda _a.\] Its empty product is one, including \(L=0\). These definitions use the specified pairing and its induced subspace volume.

For a weighted incidence Laplacian with positive real edge weights, the kernel theorem identifies \(K\) with fields constant on each connected component. Indeed its energy is a sum of positive weights times squared edge differences; zero energy means every difference is zero, and propagation along undirected paths gives those constants. Loops contribute zero and isolated vertices are components. Thus \(H\) imposes zero sum separately on each component. For a connected nonempty graph, \(n=|V|\) gives the mean-zero field space \[\begin {aligned} u_0&=\mathbf 1_V/\sqrt n,\\ H&=\{\phi :\sum _v\phi (v)=0\},\\ P_H\phi &=\phi -\frac {\sum _v\phi (v)}{n}\mathbf 1_V. \end {aligned}\] The empty graph has \(E=K=H=0\); no division by \(n\) is used there.

The kernel complement and determinant product are direct consequences of the spectral decomposition in [boyd2004convex, Section A.5.2, pp. 646-647]; the graph kernel uses the incidence-energy identity proved in Theorem [fdip-0008].

The massless four-cycle calculation makes this kernel complement and pseudodeterminant explicit.