Description
Add aij and aji for the residual from skew symmetry.
A real matrix splits into symmetric and skew-symmetric parts by pairing transposed entries. Pairwise residuals test one relationship rather than the entire matrix.
Add aij and aji for the residual from skew symmetry.
When to use this tool
Use Matrix Skew Entry Pair Residual when you need one transposed-entry decomposition or residual.
How to use the tool and how it works
- Enter aij, aji. Use the units and conventions below.
- Run the tool. Correct any format, range, or cross-field input error before interpreting the output.
- Read residual. Use the output convention below.
Add aij and aji for the residual from skew symmetry.1
| Input field | Meaning and units | Accepted input |
|---|---|---|
| aij | Aij | Required. Finite number. |
| aji | Aji | Required. Finite number. |
| Output field | Interpretation | Returned shape |
|---|---|---|
| residual | Signed residual using the convention in the calculation paragraph. | Required. Finite number. |
json
{
"input": {
"aij": 3,
"aji": -3
},
"output": {
"residual": 0
}
}Limitations and assumptions
- Zero is an exact pairwise condition only; no tolerance or whole-matrix verification is provided.
- 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
Check all relevant entries with a documented scale-aware tolerance for numerical symmetry decisions.
References
Similar or alternative tools
- Matrix Symmetric Skew Entry Split
Return ij/2 + ji/2 as symmetric and ij/2 - ji/2 as skew.
- Matrix Packed Skewsign Alias
Map a coordinate to the upper pair min(row, column), max(row, column).