Proposition. Invariants of a selected background [fdip-005J]
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  • November 1, 2026

For a selected background, put \(X=\operatorname {im}P\). The following are invariants of its orthogonal comparison class:

  • the spectrum of \(Q\), with multiplicities, and the rank of \(P\);
  • the spectrum of \(S_P\), hence its determinant and trace;
  • for each eigenvalue \(\lambda \) of \(Q\), the principal angles between \(X\) and \(E_\lambda =\ker (Q-\lambda I)\).

Here principal angles are defined by taking orthonormal inclusion matrices \(B_X,B_\lambda \) and the \(\min (\dim X,\dim E_\lambda )\) singular values of \(B_X^*B_\lambda \). These numbers lie in \([0,1]\); their arccosines, in increasing order, are the angles. This definition includes zero singular values and uses the entire eigenspace when \(\lambda \) is repeated. It is independent of the two orthonormal bases.

Proof. 

  • November 1, 2026

If \(Q'=OQO^{-1}\) and \(P'=OPO^{-1}\), then \(O(X)=X'\) and \(O(E_\lambda )=E'_\lambda \). Choosing the transported bases leaves \(B_X^*B_\lambda \) unchanged; changing either basis multiplies this matrix on the left or right by an orthogonal matrix, which does not change its singular values. These are at most one because both inclusions and orthogonal projections have operator norm at most one. Conjugation gives the spectrum of \(Q\) and rank of \(P\). The selected-law transport gives \(S_{P'}=(O|_X)S_P(O|_X)^{-1}\), proving the remaining claims.

These are consequently invariants of both graph groupoids \(\mathsf {Bg}\) and \(\mathsf {Bg}^{\pm }\). There is additional incidence information: the isomorphism class of the weighted incidence presentation with its selector, and the order of its finite automorphism group, are invariants in the chosen graph groupoid. Indeed, an arrow \(g:b\to b'\) induces the group isomorphism \(\alpha \mapsto g\alpha g^{-1}\) of automorphism groups. The directed and signed groups are different choices here; \(|\operatorname {Aut}_{\mathsf {Bg}}(b)|\) is not being asserted invariant under an arbitrary signed arrow.

The cospectral pair has an orthogonal comparison but no graph arrow. Thus even all the listed spectral and angle invariants cannot recover incidence. No completeness claim for invariants of the pair \((Q,P)\) under \(O(H)\) is made.

TauCeti's TauCeti.cosPrincipalAngles defines the singular values of the overlap operator of two finite orthonormal families as a nonincreasing sequence padded by zeros. Taking its first \(\min (\dim X,\dim E_\lambda )\) values and then arccosines gives the convention above. Its TauCeti.cosPrincipalAngles_le_one supplies the upper bound; the background-conjugacy and incidence claims are the separate arguments here.