Evaluate symmetric quadratic inequalities with matrix-based feasibility checks. Review determinants, traces, eigenvalues, and critical points. Export clean reports for faster study, validation, and review.
Use a symmetric matrix A = [[a11, a12], [a12, a22]] with q(x) = xTAx + 2bTx + c.
| a11 | a12 | a22 | b1 | b2 | c | Matrix Class | Typical q(x) ≤ 0 Outcome |
|---|---|---|---|---|---|---|---|
| 1 | 0 | 1 | 0 | 0 | -4 | Positive definite | Bounded nonempty region |
| 1 | 0 | 1 | 0 | 0 | 2 | Positive definite | No real solution |
| -2 | 0 | -1 | 0 | 0 | 1 | Negative definite | Nonempty unbounded region |
| 1 | 2 | -1 | 0 | 0 | 0 | Indefinite | Nonempty unbounded region |
| 1 | 0 | 0 | -1 | 0 | 0 | Positive semidefinite | May stay unbounded |
Quadratic form: q(x) = xTAx + 2bTx + c
Symmetric matrix: A = [[a11, a12], [a12, a22]]
Expanded form: q(x1, x2) = a11x12 + 2a12x1x2 + a22x22 + 2b1x1 + 2b2x2 + c
Trace: tr(A) = a11 + a22
Determinant: det(A) = a11a22 - a122
Eigenvalues: λ = (tr(A) ± √((a11 - a22)2 + 4a122)) / 2
Critical point when invertible: x* = -A-1b
Critical value when finite: q(x*) = c - bTA-1b
Decision idea: the sign pattern of the eigenvalues tells whether the quadratic form is positive definite, negative definite, indefinite, or semidefinite. That sign pattern controls the shape of the feasible set for q(x) ≤ 0.
This calculator analyzes a two variable quadratic form written as q(x) = xTAx + 2bTx + c. It helps you study a linear matrix inequality style condition of the form q(x) ≤ 0. The page checks the matrix structure, the eigenvalues, and the likely shape of the feasible region. It also evaluates any sample point you enter.
Quadratic inequalities depend strongly on definiteness. A positive definite matrix creates a bowl shaped surface. A negative definite matrix creates an upside down bowl. An indefinite matrix creates a saddle. These shapes decide whether the inequality has no solution, a bounded region, or an unbounded region. That is why determinant and eigenvalue checks are essential.
The calculator reports trace, determinant, eigenvalues, and a matrix class. It also finds a representative critical point when one exists. For invertible cases, that point is the stationary point x* = -A-1b. The tool then estimates feasibility for q(x) ≤ 0. This is useful in optimization, control theory, numerical methods, and advanced algebra.
Use it when you need a quick quadratic form calculator for symmetric 2 × 2 matrices. It helps with classroom work, self study, and engineering style matrix analysis. It is also useful when checking whether a quadratic constraint is empty, bounded, or globally satisfied.
This page is designed for a clean and fast workflow. It gives interpretable outputs instead of hiding the algebra. That makes it easier to verify each step. You can compare the computed matrix type with hand calculations, test points, and exported reports. For larger semidefinite programming problems, dedicated optimization solvers are still the better choice, but this calculator is excellent for learning and quick validation.
It studies a two variable symmetric quadratic inequality of the form q(x) ≤ 0. It does not solve general large scale semidefinite programs.
Quadratic forms are analyzed through symmetric matrices. Any real quadratic form can be represented by a symmetric matrix without changing its value.
It means the quadratic surface opens upward in every direction. For q(x) ≤ 0, the feasible region can be bounded or empty, depending on the minimum value.
An indefinite matrix gives a saddle shaped surface. The inequality q(x) ≤ 0 is then typically nonempty and unbounded.
It is the stationary point or a representative minimizer or maximizer. For invertible matrices, it is computed from x* = -A-1b.
Eigenvalues reveal the curvature of the quadratic form. Their sign pattern determines definiteness and strongly guides the feasibility conclusion.
Yes. Use the CSV button for spreadsheet style output and the PDF button for a clean report of the current result block.
Yes. It is useful for checking hand calculations in linear algebra, optimization, control, and quadratic form topics.
Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.