Cauchy Determinant 2x2 Calculator
Cauchy2×2matrixAij=1/(xi+yj) determinant closed form including singular repeated nodes, nonzero denominators. Explicit bounded resource/branch and actual observations or independently verified parameters; no unproved infinite assertion certified.
Description
A Cauchy matrix uses reciprocal sums of independently specified row and column nodes.
A Cauchy matrix uses reciprocal sums of independently specified row and column nodes.1
This tool computes its 2-by-2 determinant under the nonzero-denominator convention, including zero for repeated-node singular cases.
When to use Cauchy Determinant 2x2 Calculator
Use this tool to compute its 2-by-2 determinant under the nonzero-denominator convention, including zero for repeated-node singular cases.
How to use Cauchy Determinant 2x2 Calculator and how it works
- Prepare actualXNodes, actualYNodes, with the units, conventions and assumptions below.
- Enter the required fields and run Cauchy Determinant 2x2 Calculator.
- Read result.determinant using the output definitions below.
- Correct any rejected inputs and rerun; interpret the result within the stated model and its assumptions.
| Input field | Meaning | Value type | Requirement |
|---|---|---|---|
| actualXNodes | Exactly two finite x nodes, in matrix-row order. | array of number | Required; other constraints below |
| actualYNodes | Exactly two finite y nodes, in matrix-column order. | array of number | Required; other constraints below |
Require two finite x nodes and two finite y nodes. Reject any zero pairwise sum x_i+y_j, then evaluate the 2-by-2 Cauchy determinant formula. Equal nodes within either list are allowed and produce zero; no nonsingularity assertion is inferred merely from accepted input.
determinant=(x_2-x_1)*(y_2-y_1)/product_{i,j}(x_i+y_j) Apply the input ordering, units and branch conditions stated above.
| Output field | Meaning | Value type |
|---|---|---|
| result | Object containing the calculated or selected fields below. | object |
| result.determinant | Signed numerical determinant under the stated node ordering. | number |
{
"input": {
"actualXNodes": [
1,
2
],
"actualYNodes": [
3,
4
]
},
"output": {
"result": {
"determinant": 0.0016666666666666668
}
}
}Limitations and assumptions
The calculation uses the supplied observations and model inputs. It does not collect measurements, select a reference population or coefficient, or establish that the model applies to the actual case. The stated branch conditions and output units remain part of the result.
All fields require explicit values. Unexpected fields, wrong types, nonfinite numbers and sparse arrays are rejected. Alignment, bounds and supported-token rules are explained above; extremely large inputs or nonfinite results can also fail. Input errors differ from successful false comparisons, unavailable-domain flags and empty match lists.
Alternative or Complementary methods
Use 2x2 Eigenvectors Calculator for a complementary operation: A real matrix can have real, repeated or complex eigenvalues, and repeated values do not ensure diagonalizability.
Use 2x2 Condition Number Calculator for a complementary operation: A spectral condition number compares the largest and smallest singular values of a matrix.
References
-
Cauchy Determinant 2x2 Calculator — tool.io
Similar or alternative tools
- 2x2 Eigenvectors Calculator
A real matrix can have real, repeated or complex eigenvalues, and repeated values do not ensure diagonalizability.
- 2x2 Condition Number Calculator
A spectral condition number compares the largest and smallest singular values of a matrix.