Let \(L:M\to M\) be an endomorphism of a finite free module over a commutative unital ring \(k\). On the domain \(\operatorname {tr}_M(L)\in k^\times \), define the trace ratio \[R(L)=\operatorname {tr}_M(L^2)\bigl (\operatorname {tr}_M(L)\bigr )^{-2},\] using intrinsic trace. If \(P:M\to M'\) is an isomorphism, \(c\in k^\times \), and \(L'=cPLP^{-1}\), then \(\operatorname {tr}_{M'}(L')\) is a unit and \(R(L')=R(L)\).

Consider a finite-dimensional positive inner-product space over \(\mathbb R\). A nonzero self-adjoint positive-semidefinite operator on it has positive trace. Indeed an orthonormal eigenbasis gives eigenvalues \(\lambda _i\geq 0\); if all were zero the operator would be zero. Hence \(\operatorname {tr}L=\sum _i\lambda _i>0\). This verifies the denominator for a nonzero positive-weight graph Laplacian.

[boyd2004convex, Appendix A.5.2, pp. 646-647] gives the spectral and trace formulas. The exceptional examples show why nonzero alone is insufficient over other rings or without positivity.

For nonzero real positive-semidefinite Laplacians with their counting pairings, write \(L\approx L'\) when \(L'=cPLP^{-1}\) for a positive scalar and a vertex permutation. Identity uses \((1,I)\); inverse uses \((c^{-1},P^{-1})\); composition uses the products of the scalars and permutations. This is an equivalence relation with invariant \(R\). It forgets the mass, normalized state and retained partition of an enriched background. It also differs from arbitrary congruence of bilinear-form arrays: the ratio compares the associated endomorphisms. It is an invariant, not a classification assertion.

The ring calculation is a direct consequence of Definition [fdip-003G], whose trace convention is [stacks2017stacks, Exercise 22.6], Tag 02DU. Mathlib's Matrix.trace_smul supplies the coordinate scalar identity; the unit domain and cancellation above are additional hypotheses and deductions. The cycle comparison computes the invariant and the associated normalized observables explicitly.