Description
Return column dot product ab + cd and squared-norm residuals a² + c² - 1 and b² + d² - 1 for the matrix [[a,b],[c,d]].
Row margins, entry thresholds, and small-matrix criteria summarize specific algebraic conditions. They do not automatically establish a larger matrix's conditioning or spectrum.
Return column dot product ab + cd and squared-norm residuals a² + c² - 1 and b² + d² - 1 for the matrix [[a,b],[c,d]].
When to use this tool
Use Matrix 2x2 Orthogonal Column Residual when you need one explicitly scoped matrix bound or residual.
How to use the tool and how it works
- Enter a, b, c, d. Use the units and conventions below.
- Run the tool. Correct any format, range, or cross-field input error before interpreting the output.
- Read dot, firstNormResidual, secondNormResidual. Use the output convention below.
Return column dot product ab + cd and squared-norm residuals a² + c² - 1 and b² + d² - 1 for the matrix [[a,b],[c,d]].1
| Input field | Meaning and units | Accepted input |
|---|---|---|
| a | A | Required. Finite number. |
| b | B | Required. Finite number. |
| c | C | Required. Finite number. |
| d | D | Required. Finite number. |
| Output field | Interpretation | Returned shape |
|---|---|---|
| dot | Dot | Required. Finite number. |
| firstNormResidual | First Norm Residual | Required. Finite number. |
| secondNormResidual | Second Norm Residual | Required. Finite number. |
{
"input": {
"a": 1,
"b": 0,
"c": 0,
"d": 1
},
"output": {
"dot": 0,
"firstNormResidual": 0,
"secondNormResidual": 0
}
}Limitations and assumptions
- Residuals are numeric diagnostics without an acceptance tolerance; no Boolean orthogonality verdict is returned.
- 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 the complete matrix and a suitable numerical tolerance or factorization for a global property test.
References
Similar or alternative tools
- Matrix Two Row Inner Product Term
Require equal-length left and right arrays, multiply corresponding elements into terms, and sum those products.
- Matrix Symmetric Skew Entry Split
Return ij/2 + ji/2 as symmetric and ij/2 - ji/2 as skew.