Build symmetric matrices and test feasible regions. Track objective changes, export findings, and validate constraints. Understand small optimization problems through practical visual matrix analysis.
This solver searches over a symmetric 2×2 matrix X = [[x11, x12], [x12, x22]].
The default values below create a simple feasible semidefinite optimization example.
| Item | Value | Meaning |
|---|---|---|
| Objective C | [ [1, 0], [0, 1] ] | Minimizes x11 + x22, which is the trace of X. |
| Constraint 1 | [ [1, 1], [1, 1] ] • X = 4 | Enforces x11 + 2x12 + x22 = 4. |
| Constraint 2 | [ [1, 0], [0, -1] ] • X = 0 | Forces x11 = x22. |
| Bounds | x11 ∈ [0,4], x12 ∈ [-2,2], x22 ∈ [0,4] | Search interval for the grid-based solver. |
| Expected pattern | x11 ≈ 1, x12 ≈ 1, x22 ≈ 1 | Produces a boundary PSD solution with determinant near zero. |
This page solves a small semidefinite model for a symmetric decision matrix:
The objective uses the symmetric matrix inner product:
Each equality constraint uses the same structure:
The positive semidefinite condition for a 2×2 symmetric matrix is checked through:
The eigenvalues are reported with:
Because this page is self-contained, it uses an approximate grid search. It is excellent for learning, checking examples, and exploring small 2×2 semidefinite programs.
It solves a small semidefinite optimization model for a 2×2 symmetric matrix. You can minimize or maximize a linear matrix objective while enforcing equality constraints and the positive semidefinite condition.
For a 2×2 symmetric matrix, positive semidefinite means both diagonal entries are nonnegative and the determinant is nonnegative. These conditions ensure the matrix has no negative eigenvalues.
No. This page uses a grid-based numerical search. It is approximate, but very useful for demonstrations, quick feasibility checks, classroom examples, and small exploratory optimization tasks.
A grid search rarely lands on an equality exactly. Tolerance lets the page accept points that are numerically close to the required constraints and semidefinite boundary.
The matrix inner product counts both off-diagonal terms of a symmetric matrix. Since x12 appears in positions (1,2) and (2,1), the contribution becomes twice the off-diagonal coefficient.
Start with mathematically reasonable limits from your model. Narrow bounds speed the search and improve detail. Wide bounds are helpful when you do not yet know where the feasible region lies.
The chart shows feasible sampled matrices using x11, x12, and x22 as coordinates. Marker coloring follows the objective value, helping you see where stronger solutions tend to appear.
Use a dedicated optimizer for larger matrices, many constraints, exact certificates, or production research. This page is best for compact educational problems and fast visual experimentation.
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.