Study symmetric matrices, quadratic forms, and inequality conditions. Check definiteness, pivots, and vector responses instantly. Save outputs for classes, audits, reports, and revision tasks.
The quadratic form is q(x) = xTAx + 2gTx + c, where A = [[a11, a12], [a12, a22]] and x = [x1, x2]T.
For a 2×2 symmetric matrix, the eigenvalues are:
λ1,2 = (trace(A) ± √(trace(A)2 - 4det(A))) / 2
trace(A) = a11 + a22 and det(A) = a11a22 - a122.
Positive semidefinite requires λmin ≥ 0. Positive definite requires λmin > 0. Negative semidefinite requires λmax ≤ 0. Negative definite requires λmax < 0.
If det(A) ≠ 0, the stationary point solves Ax + g = 0, so x* = -A-1g.
| Case | a11 | a12 | a22 | g1 | g2 | c | Likely Class |
|---|---|---|---|---|---|---|---|
| Study Set 1 | 4 | 1 | 3 | -2 | 1 | 5 | Positive definite |
| Study Set 2 | -3 | 0 | -2 | 1 | -1 | 0 | Negative definite |
| Study Set 3 | 1 | 2 | -1 | 0 | 0 | 2 | Indefinite |
| a11 | a12 |
| a12 | a22 |
Use a symmetric matrix to keep the quadratic form consistent. The tool compares the matrix condition and the pointwise inequality at the same time.
A linear matrix inequality linked to a quadratic form helps test stability, convexity, and bounded behavior. This calculator focuses on a symmetric 2×2 matrix. That setup is common in optimization, control theory, and numerical analysis. It also fits classroom exercises and engineering checks. You enter matrix coefficients, linear terms, and a constant. The tool then evaluates the quadratic form at a chosen vector. It also tests whether the matrix satisfies positive semidefinite, positive definite, negative semidefinite, or negative definite conditions.
Definiteness explains the shape of a quadratic surface. A positive definite matrix produces a strict minimum. A negative definite matrix produces a strict maximum. A semidefinite matrix can be flat in one direction. An indefinite matrix creates a saddle pattern. These classifications matter when you study constrained optimization, Lyapunov functions, and feasibility regions. They also help you decide whether an inequality is globally satisfied or only locally meaningful.
The calculator returns the trace, determinant, eigenvalues, principal minors, Schur complement, and feasibility margin. It also solves the stationary point when the matrix is invertible. That point is found from the gradient condition. The resulting output helps verify the structure of the quadratic form quickly. Because the matrix is symmetric, the eigenvalue test and principal minor test are both useful. The tool also shows whether your selected inequality target is satisfied.
This page is useful for students, analysts, and instructors. It reduces manual algebra and lowers sign errors. It also creates neat result tables for reports, homework review, and audit notes. The export buttons help save values for later comparison. The example table shows how a valid input set behaves. Use it to compare positive, negative, and mixed curvature cases.
The quadratic value at a test vector is also important. It shows whether one specific point satisfies the inequality threshold. Combined with eigenvalues and the stationary point, that value gives both local and global insight. This makes the calculator practical for tutorials, verification, and quick decision support in class. Use the calculator when you need faster matrix screening, cleaner reporting, and a repeatable study process. It is especially helpful for homework checking, exam practice, and quick parameter sensitivity review before final submission.
An LMI here means a matrix condition such as A ≽ 0 or A ≼ 0 for the symmetric matrix inside the quadratic form. The calculator checks whether that condition holds using eigenvalues and principal minors.
A real quadratic form is naturally linked to a symmetric matrix. Symmetry guarantees real eigenvalues and makes definiteness tests reliable. If your original coefficients are not symmetric, first rewrite them into an equivalent symmetric form.
The tool uses eigenvalues, determinant, trace, and principal minors. For a 2×2 symmetric matrix, these tests agree and give a fast classification into positive definite, positive semidefinite, negative definite, negative semidefinite, or indefinite.
The feasibility margin measures how far the matrix is from meeting your selected inequality target. A positive margin means the target is satisfied with slack. A negative margin means the matrix violates that target.
A zero or near-zero determinant means the matrix may be singular. In that case, the stationary point may not be unique or computable by inversion. The calculator still reports the definiteness tests and point evaluation.
The stationary point solves the gradient equation of the quadratic form. It identifies where the form can reach a minimum, maximum, or saddle location, depending on the definiteness of the symmetric matrix.
No. This version is designed for a symmetric 2×2 matrix and an associated quadratic form. It is ideal for teaching, quick checks, and compact reports. Larger LMIs need dedicated numerical solvers.
CSV is useful for spreadsheets and audit trails. PDF is useful for printing, sharing, and keeping a clean study record. Both options make it easier to preserve results from each matrix case.
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.