Proposition. The Laplacian Moore-Penrose inverse [fdip-0032]
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  • October 29, 2026

With the kernel decomposition, define the Laplacian pseudoinverse by \[L^+=iL_H^{-1}p_H.\] This is the unique endomorphism \(A\) satisfying the 4 Moore-Penrose identities: \[\begin {aligned} LAL&=L,& ALA&=A,\\ (LA)^\dagger &=LA,&(AL)^\dagger &=AL. \end {aligned}\] It vanishes on \(K\), is the inverse of \(L_H\) on \(H\), and has spectral expression \(L^+=\sum _{\lambda _a>0}\lambda _a^{-1}P_{u_a}\), where \(P_u x=h(u,x)u\) for a unit eigenvector.

Proof. 

  • October 29, 2026

On the orthogonal decomposition \(K\mathbin \perp H\), the operators \(L,L^+\) have blocks \((0,L_H)\) and \((0,L_H^{-1})\). Hence \[LL^+=L^+L=P_H.\] The inverse on \(H\) is self-adjoint by the inverse-response lemma. Thus \(L^+\) is self-adjoint, both products are self-adjoint, and multiplication once more gives \(LL^+L=L\) and \(L^+LL^+=L^+\).

Suppose \(A\) satisfies the 4 identities. Then \(LA\) is self-adjoint and idempotent, since \(LALA=LA\). Its range lies in \(\operatorname {ran}L=H\), and \(LAL=L\) shows it acts as the identity on \(H\). Thus it is the orthogonal projection \(P_H\). Similarly \(AL\) is self-adjoint and idempotent. Its kernel is \(K\): \(Lx=0\) implies \(ALx=0\), while \(ALx=0\) implies \(Lx=LALx=0\). A self-adjoint idempotent is the orthogonal projection onto its kernel complement, so \(AL=P_H\). For \(y=Lx\in H\), \[Ay=ALx=P_Hx=L_H^{-1}y.\] Finally \(A=ALA=A(LA)=AP_H\), so \(A\) vanishes on \(K\). These two restrictions force \(A=L^+\). The argument includes \(H=0\) and \(E=0\); the unique map on the zero space is its own inverse.

TauCeti's LinearMap.IsMoorePenroseInverse uses exactly these 4 conditions for real inner-product spaces. Its LinearMap.IsMoorePenroseInverse.unique supports uniqueness, and the finite-dimensional characterization LinearMap.isMoorePenroseInverse_iff_eq_moorePenroseInverse identifies this construction with \(L.\mathrm {moorePenroseInverse}\). The weaker inner-inverse identity \(LAL=L\) alone gives neither this uniqueness nor a constrained probability law.

The construction is a direct specialization of the orthogonal spectral decomposition in [boyd2004convex, Section A.5.2, pp. 646-647]. The four-identity and uniqueness arguments above establish the additional Moore-Penrose conclusion; no Gaussian law follows from that conclusion alone.