Theorem. Partial Grassmann integration and the Schur complement [fdip-009J]
✍️source

  • November 1, 2026

Let \(R,U\) be disjoint ordered finite sets. Order the independent paired generators with all retained \(R\)-pairs first and all eliminated \(U\)-pairs second. For \(n=|R|+|U|\), let \[K=\begin {pmatrix}A&B\\C&D\end {pmatrix} \in M_n(\mathbb C).\] Require \(D=K_{UU}\) to be invertible and put \(S=A-BD^{-1}C\). No adjoint relation between \(B,C\) is assumed. Define \(J_U\) by taking the coefficient of the ordered \(U\)-pair top word after moving retained generators to the left. The result is an element of the retained exterior algebra. Define \(J_R\) by its paired top word, so \(J_RJ_U=J_n\). The eliminated top word has even degree, so moving it past a retained coefficient gives no further sign. Then \[J_U\left (e^{-\bar \psi K\psi }\right ) =\det D\,e^{-\bar \psi _R S\psi _R},\] and consequently \[\det K=\det D\det S.\] The empty-block conventions are determinant one and the identity coefficient map. Invertibility of \(K\) or \(S\) is unnecessary.

Proof. 

  • November 1, 2026

Introduce the odd linear combinations \[\begin {aligned} \chi _U&=\psi _U+D^{-1}C\psi _R,\\ \bar \chi _U&=\bar \psi _U+\bar \psi _RBD^{-1}. \end {aligned}\] The bars remain labels of independent generators. Expanding in the displayed multiplication order gives \[\bar \psi K\psi =\bar \psi _R S\psi _R+\bar \chi _U D\chi _U.\] For example, the last term of the expansion of \(\bar \chi _U D\chi _U\) is \(\bar \psi _RBD^{-1}C\psi _R\), which cancels the subtraction in \(S\). All these quadratic elements are even. They commute, so their finite exponentials multiply.

Replacing an eliminated generator by itself plus a retained odd linear combination preserves its partial top coefficient. Indeed, expand each exterior monomial in the new symbols: every term using a retained replacement has fewer eliminated generators, while the term using no replacement has coefficient one and its original order. Terms of lower eliminated degree cannot contribute to the top degree. Thus the coefficient in the shifted quadratic exponential equals the coefficient for independent \(U\)-generators. The paired determinant identity evaluates it as \(\det D\). The retained exponential factors on the left, proving the partial formula. Applying \(J_R\) and the same determinant identity proves the full factorization.

The effective matrix is the same block expression as in real positive Gaussian elimination. The scalar here is \(\det D\); the real Gaussian fiber integral instead has \((2\pi )^{|U|/2}(\det D)^{-1/2}\) under its real positive hypotheses and volume convention. Neither positivity nor an ordinary integration measure was used in the Grassmann proof.

The finite block identity is also supplied by Matrix.det_fromBlocks₂₂, for finite index types over a commutative ring with an invertible eliminated block. This formal statement supports the determinant factorization, not the partial Grassmann coefficient operation.