Theorem. When a marginal precision is a restriction [fdip-005M]
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  • November 1, 2026

Let \(Q>0\) be self-adjoint on a finite-dimensional real Euclidean space \(H\) and let \(P\) be an orthogonal projector. Assume its image \(X\) and kernel \(Y\) are nonzero. In \(H=X\oplus Y\), write \[Q=\begin {pmatrix}A&K\\K^*&D\end {pmatrix},\qquad D>0.\] The selected precision obeys \[S_P=A-KD^{-1}K^*\preceq A.\] Here \(A=PQ|_X\) is the compression, and \(T\preceq A\) means that \(\langle x,(A-T)x\rangle \ge 0\) for every \(x\in X\). The following conditions are equivalent: \[\begin {aligned} S_P=A&\quad \Longleftrightarrow \quad K=0\\ &\quad \Longleftrightarrow \quad QX\subseteq X \quad \Longleftrightarrow \quad PQ=QP. \end {aligned}\] In this case \(A=Q|_X\); call the selector restricting. Otherwise it induces a different precision through elimination of the complement.

Proof. 

  • November 1, 2026

The Schur formula is proved in Theorem [fdip-005B]. Its difference from the compression satisfies \[\langle x,(A-S_P)x\rangle =\|D^{-1/2}K^*x\|^2\ge 0.\] It vanishes for all \(x\) exactly when \(K^*=0\), since \(D^{-1/2}\) is invertible. The \(Y\)-component of \(Q(x,0)\) is \(K^*x\), so the same condition is \(QX\subseteq X\). Self-adjointness then makes \(Y=X^\perp \) invariant as well: \(\langle Qy,x\rangle =\langle y,Qx\rangle =0\). The resulting block diagonal matrix commutes with \(P\); conversely \(PQ=QP\) carries its image to itself.

When \(K\ne 0\), the positive semidefinite difference \(A-S_P\) is nonzero, so the quadratic form decreases in some direction. It need not be positive definite: some directions may have zero loss. The assertion is a Loewner inequality with the stated equality criterion, not a strict inequality in every direction.

For a positive-weight graph and a coordinate selector \(P_R\), \(K\) consists of the entries of \(Q\) between \(R\) and its complement. For distinct vertices these are minus the sum of weights of edges joining them; orientation does not affect the sum. All summands are positive. Hence \(K=0\) exactly when no edge joins \(R\) to its complement. In a connected graph every nonempty proper coordinate selector therefore induces a different precision. In particular, the retained vertices in cycle decimation are stable under the retained cyclic action, but this symmetry does not make their span \(Q\)-invariant.

In orthonormal coordinates, Mathlib's Matrix.PosDef.inv gives positivity of \(D^{-1}\). The congruence lemma Matrix.PosSemidef.mul_mul_conjTranspose_same, with its positive semidefinite matrix set to \(D^{-1}\) and its rectangular matrix set to \(K\), gives \(KD^{-1}K^*\succeq 0\). These declarations support the positivity steps. The equality criterion, subspace equivalences and graph specialization are proved above.