Solve eigenvalues for 2×2, 3×3, and larger matrices. Inspect trace, determinant, and stability measures quickly. Plot spectral magnitude patterns for clear matrix interpretation today.
| Example | Matrix | Expected Insight |
|---|---|---|
| Symmetric 2×2 | [2 1; 1 2] | Real eigenvalues with exact closed-form verification. |
| Tridiagonal 3×3 | [4 1 0; 1 3 1; 0 1 2] | Stable real spectrum and clean QR convergence. |
| Dense 4×4 | [6 2 1 0; 2 5 2 1; 1 2 4 2; 0 1 2 3] | Useful for comparing spectral radius and trace. |
Eigenvalues λ satisfy the characteristic equation: det(A − λI) = 0.
For a 2×2 matrix A = [a b; c d], the characteristic polynomial is λ² − (a + d)λ + (ad − bc) = 0.
This calculator uses shifted QR iteration for numerical approximation. Repeated factorization writes Aₖ = QₖRₖ and updates Aₖ₊₁ = RₖQₖ. The diagonal entries of the nearly upper triangular matrix approach the eigenvalues.
Supporting checks: trace(A) = Σλᵢ, det(A) = Πλᵢ, and spectral radius = max |λᵢ|.
It computes approximate eigenvalues of a square matrix, along with trace, determinant, Frobenius norm, spectral radius, Gershgorin bounds, and QR convergence data.
Yes. Eigenvalues are defined for square matrices only. Every row must contain the same number of entries as the total number of rows.
For 2×2 matrices, the page also shows an exact closed-form check. Larger matrices use numerical QR iteration, so results are approximations.
They are fast validation checks. The sum of eigenvalues should match the trace, and the product should match the determinant within numerical tolerance.
The spectral radius is the largest absolute eigenvalue. It is useful in stability analysis, iterative methods, and dynamic systems.
QR convergence describes how quickly off-diagonal terms shrink during iteration. Smaller final off-diagonal norms indicate a more reliable triangular approximation.
The exact 2×2 routine reports complex conjugates when needed. For larger matrices, this implementation mainly targets real-valued approximations from QR iteration.
Place one row on each line. Separate entries using spaces, commas, or semicolons. Decimal and negative values are supported.
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.