Corollary. Positive multiplicity blocks and real dimensions [fdip-0050]
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  • November 1, 2026

Under the commutant identification, an equivariant self-adjoint positive definite precision \(Q\) is specified by \(Q_\Xi \in M_{m_\Xi }(D_\Xi )\) with \(Q_\Xi ^*=Q_\Xi >0\). Thus all such precisions form an open cone in a real vector space of dimension \[\sum _\Xi h(m_\Xi ,D_\Xi ),\qquad \begin {array}{c|ccc} D&\mathbb R&\mathbb C&\mathbb H\\\hline h(m,D)&m(m+1)/2&m^2&2m^2-m. \end {array}\] For a vertex permutation representation this dimension also counts \(G\)-orbits on unordered pairs \(\{u,v\}\), including \(u=v\). This is the full cone of equivariant precisions; graph Laplacians with prescribed edges and positive weights generally form a smaller family.

Proof. 

  • November 1, 2026

Choose an orthonormal basis of \(W_\Xi \) as a left \(D_\Xi \)-space, using the positive adjoint of Theorem [fdip-004Z]. Explicitly, an orthonormal real set \(e,ie\) or \(e,ie,je,ke\) spans one such scalar line; its real orthogonal complement is \(D_\Xi \)-stable, so induction constructs the basis. Put \(r_\Xi =\dim _{D_\Xi }W_\Xi \) and \(d_D=\dim _{\mathbb R}D\in \{1,2,4\}\). Expanding each of the \(m_\Xi \) vectors in this basis identifies \(H_\Xi \) with \(r_\Xi \) columns in \(D_\Xi ^{m_\Xi }\). The same matrix \(Q_\Xi \) acts on every column by left multiplication. The real inner product is the sum of \(\operatorname {Re}(z^*w)\) on these columns. Consequently positivity of \(Q|_{H_\Xi }\) is exactly positivity of \(Q_\Xi \).

A Hermitian matrix has \(m\) real diagonal entries and \(m(m-1)/2\) freely chosen off-diagonal entries in \(D\); the opposite entries are their conjugates. This gives \(h(m,D)=m+d_Dm(m-1)/2\). Positive definiteness is open because \(\langle Qx,x\rangle \) has a strictly positive minimum on the unit sphere. In permutation coordinates a symmetric commuting matrix is constant on unordered-pair orbits, whose indicator matrices again form a basis.

These coordinates fix the determinant convention without a noncommutative determinant ambiguity: \[\begin {aligned} \dim _{\mathbb R}H_\Xi &=m_\Xi r_\Xi d_{D_\Xi },\\ \det _{\mathbb R}(Q|_{H_\Xi }) &=\det _{\mathbb R}(Q_\Xi :D_\Xi ^{m_\Xi }\to D_\Xi ^{m_\Xi })^{r_\Xi }. \end {aligned}\] For a positive Hermitian block with eigenvalues \(\lambda _1,\ldots ,\lambda _m\), the latter real determinant is \(\prod _j\lambda _j^{d_D}\). The spectral assertion follows by taking an eigenvector of the real self-adjoint operator: its eigenspace is stable under right scalar multiplication on \(D^m\); its scalar line has an invariant orthogonal complement, and induction gives a \(D\)-orthonormal eigenbasis. In particular a complex determinant is squared, and the product of quaternionic Hermitian eigenvalues is raised to the fourth power, before taking the exponent \(r_\Xi \). This is the positive-matrix specialization of the real block classification in [serre1977linear, Section 13.2, pp. 108-109].

For a graph write \(L=d^*Wd\) and \(Q=L+m^2I\), with \(m^2>0\). The multiplicity matrices obey \(Q_\Xi =L_\Xi +m^2I\). A zero Laplacian eigenvalue therefore gives a strictly positive precision eigenvalue \(m^2\), not a normalized massless Gaussian.