Description
Calculate a12 × b21 - b12 × a21 for the first diagonal entry of AB - BA and return its negative for the second.
Matrix-product entries are dot products of matching rows and columns. Trace and Rayleigh-quotient contributions require corresponding coordinates from the same underlying calculation.
Calculate a12 × b21 - b12 × a21 for the first diagonal entry of AB - BA and return its negative for the second.
When to use this tool
Use Matrix 2x2 Commutator Diagonal Entry when you need one explicit real matrix-product contribution.
How to use the tool and how it works
- Enter a12, a21, b12, b21. Use the units and conventions below.
- Run the tool. Correct any format, range, or cross-field input error before interpreting the output.
- Read first, second. Use the output convention below.
Calculate a12 × b21 - b12 × a21 for the first diagonal entry of AB - BA and return its negative for the second.1
| Input field | Meaning and units | Accepted input |
|---|---|---|
| a12 | A12 | Required. Finite number. |
| a21 | A21 | Required. Finite number. |
| b12 | B12 | Required. Finite number. |
| b21 | B21 | Required. Finite number. |
| Output field | Interpretation | Returned shape |
|---|---|---|
| first | First | Required. Finite number. |
| second | Second | Required. Finite number. |
{
"input": {
"a12": 2,
"a21": 3,
"b12": 4,
"b21": 5
},
"output": {
"first": -2,
"second": 2
}
}Limitations and assumptions
- This uses the two-by-two diagonal cancellation identity; off-diagonal commutator entries and whole-matrix commutativity are not evaluated.
- 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
Validate complete shapes and coordinate correspondence before aggregating contributions or using a matrix library.
References
Similar or alternative tools
- Matrix 2x2 Orthogonal Column Residual
Return column dot product ab + cd and squared-norm residuals a² + c² - 1 and b² + d² - 1 for the matrix [[a,b],[c,d]].
- Matrix 2x2 Cayley Hamilton Residual
Compute trace and determinant, then return the four row-major entries of A² - trace(A)A + det(A)I.