Schur Complement and Kron Reduction
Node role: Method.
Reader task: Eliminate declared network or spatial coordinates from an admittance, static Maxwell-capacitance, or open dynamic model while preserving their loading, response maps, and pole meaning at the retained coordinates.
Mathematical Role And Physical Boundary
Multiport simulations often expose more ports or modes than a design decision needs. Deleting their rows and columns discards coupled feedback. Kron reduction instead absorbs that feedback into an equivalent admittance seen at the retained coordinates.
The Schur complement is the mathematical elimination tool. The circuit model still decides which coordinates may be eliminated and which physical boundary condition they obey.
Physical basis
Physical Circuit Coordinates and Open-EOM Reduction defines the coordinate map, complete open operator, retained set, and port maps before this method eliminates anything. This page adds the Schur/Kron operation; it does not redefine the circuit physics.
Core model
The intuition is feedback through the eliminated coordinates: their voltages still respond to the retained ones even without an independent injected current. Kron network reduction provides the starting zero-injection boundary (D"orfler and Bullo 2013). The following block solve is this page’s derivation; its later dynamic and electrostatic extensions explicitly retain their own assumptions.
Partition a frequency-dependent admittance matrix into retained coordinates \(k\) and eliminated coordinates \(d\):
\[ \begin{bmatrix} \mathbf i_k \\ \mathbf i_d \end{bmatrix} = \begin{bmatrix} Y_{kk} & Y_{kd} \\ Y_{dk} & Y_{dd} \end{bmatrix} \begin{bmatrix} \mathbf v_k \\ \mathbf v_d \end{bmatrix}. \]
Kron reduction uses the boundary condition that there is no independent external current injection at the eliminated coordinates:
\[ \mathbf i_d=0. \]
Solving the eliminated block gives
\[ \mathbf v_d=-Y_{dd}^{-1}Y_{dk}\mathbf v_k, \]
and therefore
\[ \boxed{Y_{\mathrm{eff}} =Y_{kk}-Y_{kd}Y_{dd}^{-1}Y_{dk}}. \]
The second term is why Kron reduction is not row/column deletion: it preserves the eliminated coordinates’ coupled response as seen from the retained set.
Dynamic spatial model-order reduction
Here the starting operand is the complete open operator derived in the node/line-flux bridge, not only a static graph admittance. Apply the same block substitution to both operator and response maps. This is this page’s linear algebra; the selected coordinate names are an example, not a fixed topology required by Kron theory.
A distributed/lumped open circuit starts from its complete ordered dynamic operator, not from a pre-existing finite Equivalent Circuit. For a memoryless port boundary,
\[ \mathbf D_{\mathrm{hyb}}(\omega) =\mathbf K-\omega^2\mathbf C-i\omega\mathbf G_P. \]
First perform every physical coordinate operation required to define the model: choose the reference conductor, transform the physical node-flux coordinates, and remove a declared neutral or gauge sector. Those operations define the coordinate basis. They are not the dynamic elimination below.
Let the physically anchored retained coordinates be
\[ A=(q,r,p,f_1,f_c,f_2), \]
and let \(I\) contain every remaining spatial or internal coordinate in the compiled open basis. Wherever \(\mathbf D_{II}(\omega)\) is nonsingular, the exact retained operator is
\[ \boxed{ \mathbf D_A^{I\downarrow}(\omega) =\mathbf D_{AA}(\omega) -\mathbf D_{AI}(\omega)\mathbf D_{II}^{-1}(\omega) \mathbf D_{IA}(\omega). } \]
This operator is generally frequency dependent. The same block solve must be applied to the drive and observation maps, so the retained response agrees with the complete model. The operation is an exact viewpoint of that model; it does not produce a fixed-\(LC\) Equivalent Circuit.
For a quarter-wave resonator, the grounded head is the fixed reference boundary, the open-tail terminal is the retained resonator coordinate, and the internal \(\pi\)-section nodes belong to \(I\). For a coupled readout/filter MTL pair, eliminate the two lines’ internal coordinates jointly. Reducing each line independently and inventing a coupling afterward changes the operator and is forbidden.
Nested reductions and objective coordinates
After the spatial reduction, a target may retain only \(R=(r,p)\) and dynamically eliminate \((q,f_1,f_c,f_2)\). Equivalently, it may retain \(R\) directly from the complete Hybridized basis and eliminate the full complement. These two routes are algebraically equivalent only when every eliminated block is nonsingular and the transformed drive/observation maps, root branches, and eliminated-pole accounting are handled consistently.
Local roots, slopes, and bridge-response matching may later derive a finite set of winner-only \(LC\) readbacks. Those values are outputs of a separate response matching operation, not outputs of the Schur complement itself. They cannot replace the exact dynamic reduction inside an optimization loop.
In the earlier scoped extraction workflow, an eliminated-block singularity or a hidden eliminated-sector pole in its review window made the requested quantity NOT_EVALUABLE; static condensation or a frequency-independent \(LC\) substitution was not an authorized fallback. This is an extraction policy, not a general Schur theorem or a newly activated Gate. The block formula is pointwise valid wherever its eliminated block is nonsingular. A singular eliminated block need not be a pole visible in every full-system response; full-operator and port-projection accounting determine that interpretation.
Anchored-bare diagonal roots and hybridized poles
The retained coordinates are physical declarations, not spectral labels. If \(R=(r,p)\) is fixed by node selectors or projections before reduction, the complex diagonal roots
\[ [\mathbf D_R^{E\downarrow}(\widetilde\omega_{r,\mathrm{AB}})]_{rr}=0, \qquad [\mathbf D_R^{E\downarrow}(\widetilde\omega_{p,\mathrm{AB}})]_{pp}=0 \]
are the anchored-bare open-EOM parameters. They include loading and memory from \(E\) and the attached environment, but they are bare relative to the mutual coupling still present between retained \(r\) and \(p\). For a retained sub-block, use the corresponding diagonal-block determinant/root problem.
Residue or slope normalization converts the raw retained off-diagonal to the local open-EOM coupling \(J_{\rm res}(\omega_*)\). Solving
\[ \det\mathbf D_R^{E\downarrow}(\widetilde\omega_{\mu,\mathrm H})=0 \]
instead yields hybridized poles, the singularities of the retained susceptibility \((\mathbf D_R^{E\downarrow})^{-1}\). A hybridized pole cannot be used to rename or reassign an anchored-bare diagonal root. Root identity comes from the declared retained coordinate and branch continuation, never from frequency sorting.
The reusable vocabulary and linewidth convention are owned by Anchored-Bare Open-EOM and Hybridized Readout–Filter Parameters.
Static Maxwell-capacitance specialization
Voltage-to-charge response has the same block structure, but the physical boundary is zero supplied charge rather than zero current. This page derives the static specialization by that substitution. It retains polarization and does not confer physical ownership of a diagonal already modeled elsewhere.
The same block solve applies to a static Maxwell capacitance matrix \(\mathbf Q=\mathbf C_{\mathrm{Maxwell}}\mathbf V\) after the declared reference conductor is fixed. For electrically disconnected floating conductors \(f\), the physical boundary is zero externally supplied charge, \(\mathbf Q_f=0\). Retaining conductors \(r\) therefore gives
\[ \boxed{ \mathbf C_{\mathrm{Maxwell,red}} =\mathbf C_{\mathrm{Maxwell},rr} -\mathbf C_{\mathrm{Maxwell},rf} (\mathbf C_{\mathrm{Maxwell},ff})^{-1} \mathbf C_{\mathrm{Maxwell},fr}} \]
where implementations solve the \(ff\) block rather than forming its inverse. This operation retains the electrostatic polarization of disconnected metal; deleting its rows and columns does not.
The reduced matrix is still a Maxwell matrix, not a scalar capacitance. Its off-diagonal entries define retained mutual branches and its row sums define retained reference shunts. A later differential-mode reduction may derive a scalar \(C_{\mathrm{eff}}\), but that scalar alone cannot preserve coupling to a third retained conductor.
If one retained conductor is already represented by a distributed component, the artifact contract must say which diagonal/self-capacitance that component owns. Kron reduction does not authorize adding the retained diagonal again as a lumped shunt. Doing so double-counts stored electric energy.
What the boundary condition means
\(\mathbf i_d=0\) means no external injection at the eliminated coordinates. It does not mean that every physical load there is removed. A physical shunt or environment admittance that should remain part of the experiment must already be included in the relevant block of \(Y\) before reduction.
Zero-current Schur versus matched-wave submatrices
Use wave definitions (Kurokawa 1965) to read the omitted incident-wave condition, then compare it with the terminal-current boundary. The distinction below is this page’s derivation in the two representations, not a rule that all matrix subblocks impose the same physical load.
Zero-current elimination is not obtained by taking a submatrix of a native scattering matrix. Partition a complete scattering relation into retained ports \(k\) and omitted ports \(d\):
\[ \begin{bmatrix} b_k\\ b_d \end{bmatrix} = \begin{bmatrix} S_{kk}&S_{kd}\\ S_{dk}&S_{dd} \end{bmatrix} \begin{bmatrix} a_k\\ a_d \end{bmatrix}. \]
Reading \(S_{kk}\) sets \(a_d=0\). The omitted ports are therefore terminated in their declared matched wave references. Kron reduction instead sets the external terminal currents \(i_d=0\) while preserving every physical load already present in the network operator. These are different boundary conditions.
The matrix operation depends on the representation. In an admittance partition, \(i_d=0\) gives
\[ \boxed{Y_{\mathrm{eff}} =Y_{kk}-Y_{kd}Y_{dd}^{-1}Y_{dk}}. \]
In an impedance partition, the same boundary instead gives
\[ \boxed{Z_{\mathrm{eff}}=Z_{kk}}. \]
The Schur complement \(Z_{kk}-Z_{kd}Z_{dd}^{-1}Z_{dk}\) imposes \(V_d=0\), not \(i_d=0\). Therefore establish the physical boundary in its correct representation on the complete network, perform that representation-dependent operation, and only then convert the retained network to \(S\). A coincidental equality with \(S_{kk}\) is topology- and reference-dependent evidence; it is not a general identity. The wave/reference conversion is owned by Port Reference Impedance Semantics.
This distinction separates three operations:
- identify which explicit solver or probe shunts, if any, should be removed;
- transform into the physical port or mode coordinates, with declared normalization; and
- eliminate the coordinates whose external injections are set to zero.
The first two operations may be reordered only when every network and load matrix is transformed consistently.
Numerical implementation
A linear solve evaluates the Schur term without explicitly storing an inverse. This is an engineering implementation choice. Conditioning depends on scale and precision; it is evidence to interpret, not a universal threshold supplied by network reduction theory.
Do not form \(Y_{dd}^{-1}\) explicitly. Use
\[ Y_{dd}X=Y_{dk}, \qquad Y_{\mathrm{eff}}=Y_{kk}-Y_{kd}X. \]
An exactly singular or non-finite \(Y_{dd}\) is not permission to return a plausible trace: fail with the frequency and eliminated coordinate set.
The same rule applies to a dynamic block: solve \(\mathbf D_{II}X=\mathbf D_{IA}\) and form \(\mathbf D_A^{I\downarrow}=\mathbf D_{AA}-\mathbf D_{AI}X\). Within the same scoped extraction policy, a hidden eliminated-sector pole or failed full-operator determinant/pole closure was treated as a semantic failure even when a linear solver returned finite numbers. That policy does not change the pointwise block identity or prove that an eliminated-sector singularity is observable at every port.
A finite condition number is evidence, not an automatic approval. Its meaning depends on coordinate scaling, floating-point precision, downstream error tolerance, and the design decision being made. The implementation should publish the condition number over the sweep, but a Human reviewer chooses the acceptable threshold and explains that choice. An Agent must not hide this decision in a default constant.
Conditions for a valid reduction
- Are retained and eliminated labels explicit and stable?
- Is the input operator type (\(Y\), static Maxwell \(C\), or dynamic \(D\)) and its declared coordinate basis explicit?
- Does \(\mathbf i_d=0\) match the intended external boundary condition?
- Are physical loads that should remain already present in \(Y\)?
- Were artificial shunts removed only with artifact evidence?
- Does the implementation use a linear solve and fail visibly on singular blocks?
- Are results checked against the unreduced network at retained coordinates?
Limits and failure modes
- Applying the same block formula directly to \(S\) without a wave/reference contract is not generally Kron reduction of an admittance network.
- Eliminating a port that still has an independent source violates the \(\mathbf i_d=0\) boundary.
- Removing a physical load before reduction changes the circuit being modeled.
- Reducing in the raw port basis when the target is a differential mode can eliminate the wrong physical coordinate.
- Near a singular \(Y_{dd}\), numerical sensitivity can dominate the reported effective admittance.
- A finite result or a small algebraic reconstruction residual does not decide whether the conditioning is acceptable for a later design decision.
Connections
- Admittance Coordinate Transforms defines the physical basis and normalization used before modal reduction.
- Network Trace Views defines port order, units, and provenance for the input matrix.
- Port-Termination Compensation distinguishes conditional shunt removal from retained physical loading.
This retained engineering handoff record belongs to the earlier workflow. Its conditioning/pole-accounting fields do not create universal numeric Gates; current package schemas and any use-specific thresholds remain package/design contracts, not new requirements introduced by this explanation.
| Required field | Meaning |
|---|---|
| Input quantity | The exact input operator kind (\(Y\), static Maxwell \(C\), or open dynamic \(D\)), units, frequency axis when applicable, source artifact, and complete ordered coordinate labels. |
| Partition | Ordered retained labels and eliminated labels; index order alone is insufficient. |
| Boundary | Explicit physical boundary and, for \(Y\), whether eliminated coordinates have zero external current injection. |
| Retained loading | Physical and environmental loads left inside the operator, including loads attached to eliminated coordinates. |
| Response maps | Input/drive and output/observation maps before and after elimination, with their basis and normalization. |
| Solve policy | Linear solve method and visible exact-singular or non-finite failure context. |
| Pole accounting | Eliminated-block pole scan, retained-root branch identity, and full-operator determinant/pole closure for a dynamic reduction. |
| Conditioning evidence | Condition number at every frequency, its range and worst frequency, and the precision and coordinate basis used. |
| Human threshold decision | Accepted condition threshold, intended downstream use, rationale, reviewer, and disposition; no Agent-selected default. |
| Validation | Full-system zero-injection residual and retained-current agreement for \(Y\), or retained response plus determinant/pole closure for dynamic \(D\). |
| Lineage | Input quantity/source kind, keep/drop partition, reduction operation, output quantity/source kind, and implementation revision or working-tree state. |
References
- F. Dörfler and F. Bullo, “Kron Reduction of Graphs with Applications to Electrical Networks,” IEEE Transactions on Circuits and Systems I 60(1), 150–163 (2013), arXiv:1102.2950, doi:10.1109/TCSI.2012.2215780.
| Field | Value |
|---|---|
| Status | Seed |
| Semantic role | Define the zero-external-injection boundary, block solve, conditioning evidence, and failure modes of Kron reduction. |
| Used by | Network Trace Views, Qubit Charging-Energy Targeting and Local-System Reduction, and circuit-specific reduction workflows. |
| Implementation boundary | Child repositories own executable helpers, tests, notebook evidence, and workflow-specific conditioning thresholds. |
| Human review gate | Can the reviewer explain \(\mathbf i_d=0\), identify every retained load, reproduce the full-system equivalence check, inspect the condition range and worst frequency, and choose a threshold appropriate to the intended design decision? |