Description
For the real symmetric matrix with entries a, b, b, d, compute determinant = ad - b².
Row margins, entry thresholds, and small-matrix criteria summarize specific algebraic conditions. They do not automatically establish a larger matrix's conditioning or spectrum.
For the real symmetric matrix with entries a, b, b, d, compute determinant = ad - b².
When to use this tool
Use Matrix 2x2 Positive Semidefinite Principal Gap when you need one explicitly scoped matrix bound or residual.
How to use the tool and how it works
- Enter a, b, d. Use the units and conventions below.
- Run the tool. Correct any format, range, or cross-field input error before interpreting the output.
- Read determinant, positiveSemidefinite. Inspect positiveSemidefinite before interpreting the associated values.
For the real symmetric matrix with entries a, b, b, d, compute determinant = ad - b². Mark positiveSemidefinite when both diagonal values and determinant are nonnegative.1
| Input field | Meaning and units | Accepted input |
|---|---|---|
| a | A | Required. Finite number. |
| b | B | Required. Finite number. |
| d | D | Required. Finite number. |
| Output field | Interpretation | Returned shape |
|---|---|---|
| determinant | Determinant | Required. Finite number. |
| positiveSemidefinite | Positive Semidefinite | Required. Boolean. |
{
"input": {
"a": 2,
"b": 1,
"d": 3
},
"output": {
"determinant": 5,
"positiveSemidefinite": true
}
}Limitations and assumptions
- This exact two-by-two criterion uses no numerical tolerance; it does not test a nonsymmetric or larger matrix.
- 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 2x2 Schur Complement Scalar
Require a nonzero a and return d - c × b / a for the scalar Schur complement.
- Matrix Repeated Eigenvalue 2x2 Gap
Calculate discriminant = (a - d)^2 + 4bc.