Calculator Form
Choose a matrix size, enter the matrix values, and calculate eigenvalues for 2 × 2 or 3 × 3 square matrices.
Example Data Table
| Example | Matrix | Characteristic Polynomial | Eigenvalues |
|---|---|---|---|
| 2 × 2 Example | [5, 2] [2, 5] |
λ^2 - 10λ + 21 = 0 | 7, 3 |
| 3 × 3 Example | [4, 1, 0] [0, 2, 0] [0, 0, 1] |
λ^3 - 7λ^2 + 14λ - 8 = 0 | 4, 2, 1 |
Formula Used
- Core equation: eigenvalues are the roots of
det(λI - A) = 0. - For a 2 × 2 matrix A = [[a, b], [c, d]], the characteristic polynomial is
λ^2 - (a + d)λ + (ad - bc) = 0. - For a 3 × 3 matrix, the polynomial is
λ^3 - (tr A)λ^2 + c2λ - det(A) = 0. - Second coefficient:
c2 = ((tr A)^2 - tr(A^2)) / 2. - Checks: the sum of eigenvalues should match the trace, and the product should match the determinant.
- Graph: each eigenvalue is plotted as a point on the complex plane using its real and imaginary parts.
How to Use This Calculator
- Select whether your square matrix is 2 × 2 or 3 × 3.
- Enter the matrix values in the corresponding cells.
- Choose how many decimal places you want in the output.
- Click Find Eigenvalues to generate results above the form.
- Review the trace, determinant, characteristic polynomial, eigenvalues, and spectral radius.
- Read the complex-plane graph to see whether values are real or complex.
- Use the CSV or PDF buttons to export the result block.
Frequently Asked Questions
1. What does this calculator compute?
It computes eigenvalues for 2 × 2 and 3 × 3 square matrices. It also shows the characteristic polynomial, trace, determinant, spectral radius, verification checks, and a complex-plane graph.
2. Why do some eigenvalues contain an imaginary part?
Some real matrices naturally produce complex eigenvalues. That usually happens when the characteristic polynomial has no full set of real roots. The graph helps visualize those complex values clearly.
3. Does the matrix have to be square?
Yes. Eigenvalues are defined for square matrices only. This calculator is designed specifically for 2 × 2 and 3 × 3 square matrices.
4. Are the results exact or approximate?
The displayed values are numerical approximations, rounded to your selected decimal places. The calculator solves the characteristic polynomial numerically, which is practical for quick analysis and export.
5. What is the spectral radius?
The spectral radius is the largest magnitude among the eigenvalues. It is useful in stability checks, iteration analysis, convergence studies, and matrix power behavior.
6. Why are trace and determinant shown?
They act as quick consistency checks. The sum of eigenvalues should match the trace, and the product of eigenvalues should match the determinant, up to rounding.
7. How should I read the graph?
Each point represents one eigenvalue. The horizontal axis shows the real part, and the vertical axis shows the imaginary part. Points on the horizontal axis are purely real values.
8. Can I export my results?
Yes. After calculating, use the CSV button for spreadsheet-friendly output or the PDF button to save a printable summary of the result section.