Description
When pivot is nonzero, return corner - columnCoupling × rowCoupling / pivot.
A local matrix update changes a single entry or algebraic component under stated assumptions. Trace, squared norm, and Schur terms each respond to different structures.
When pivot is nonzero, return corner - columnCoupling × rowCoupling / pivot.
When to use this tool
Use Matrix Schur Pivot Zero Witness when you need one transparent matrix-update component.
How to use the tool and how it works
- Enter pivot, rowCoupling, columnCoupling, corner. Use the units and conventions below.
- Run the tool. Correct any format, range, or cross-field input error before interpreting the output.
- Read schurDefined, schurValue, obstructed. Inspect schurDefined, obstructed before interpreting the associated values.
When pivot is nonzero, return corner - columnCoupling × rowCoupling / pivot. At zero pivot return schurValue zero and schurDefined false; obstructed also requires a nonzero coupling.1
| Input field | Meaning and units | Accepted input |
|---|---|---|
| pivot | Pivot | Required. Finite number. |
| rowCoupling | Row Coupling | Required. Finite number. |
| columnCoupling | Column Coupling | Required. Finite number. |
| corner | Corner | Required. Finite number. |
| Output field | Interpretation | Returned shape |
|---|---|---|
| schurDefined | Schur Defined | Required. Boolean. |
| schurValue | Schur Value | Required. Finite number. |
| obstructed | Obstructed | Required. Boolean. |
{
"input": {
"pivot": 2,
"rowCoupling": 3,
"columnCoupling": 4,
"corner": 10
},
"output": {
"schurDefined": true,
"schurValue": 4,
"obstructed": false
}
}Limitations and assumptions
- The undefined zero is a placeholder. An unobstructed zero pivot does not by itself establish a generalized Schur complement or matrix invertibility.
- Inputs must satisfy both the field types and the cross-field conditions above. Invalid input causes an error; a successful false flag or documented placeholder is a different result.
- Numeric calculations use binary floating point unless exact decimal-integer arithmetic is explicitly stated. No hidden measurement tolerance or decimal rounding is added; nonfinite or out-of-contract results are rejected. Numeric signed-zero results are exported as zero.
Alternative or Complementary methods
Use a full matrix operation when validating shape, factor correspondence, pivot choices, or global numerical properties.
References
Similar or alternative tools
- Matrix 2x2 Schur Complement Scalar
Require a nonzero a and return d - c × b / a for the scalar Schur complement.
- Matrix 2x2 Positive Semidefinite Principal Gap
For the real symmetric matrix with entries a, b, b, d, compute determinant = ad - b².