Enter matrix details
Use a real symmetric matrix only. This tool accepts sizes from 2×2 through 6×6.
Example data table
| Example | Matrix | Smallest eigenvalue | Interpretation |
|---|---|---|---|
| 3 × 3 sample | [[4, 1, 1], [1, 3, 0], [1, 0, 2]] | 1.467911 | Positive definite with a moderate spectral spread. |
| 2 × 2 sample | [[2, -1], [-1, 2]] | 1.000000 | Stable symmetric system with clear separation. |
| 4 × 4 sample | [[5, 1, 0.5, 0], [1, 6, 1, 0.5], [0.5, 1, 7, 1], [0, 0.5, 1, 8]] | 4.361623 | All eigenvalues remain positive and well ordered. |
Formula used
The calculator targets real symmetric matrices because their eigenvalues are real and sortable. The smallest eigenvalue is defined as:
λmin = min eig(A)
For symmetric matrices, the same value is also the minimum Rayleigh quotient:
λmin = minx ≠ 0 (xT A x) / (xT x)
The solver applies the Jacobi rotation method. It repeatedly zeroes the largest off-diagonal term, updates the diagonal entries, and sorts the final diagonal values as eigenvalues. The reported determinant is the product of eigenvalues, and the Gershgorin lower bound estimates a guaranteed lower limit for the spectrum.
How to use this calculator
- Select a matrix size from 2×2 to 6×6.
- Enter a real symmetric matrix into the input grid.
- Adjust tolerance, decimal places, and iteration limit if needed.
- Use the mirror button to copy upper entries below the diagonal.
- Press the calculate button to show results above the form.
- Review the minimum eigenvalue, spectrum, matrix class, and eigenvector.
- Use the CSV or PDF buttons to save your report.
FAQs
1. Why does this calculator require a symmetric matrix?
Real symmetric matrices always have real eigenvalues and orthogonal eigenvectors. That makes the smallest eigenvalue meaningful, sortable, and stable for numerical reporting.
2. What does a positive smallest eigenvalue mean?
It means every eigenvalue is positive when the matrix is symmetric and positive definite. In many applications, that suggests stability, convexity, or positive energy curvature.
3. What if the smallest eigenvalue is zero?
A zero or near-zero minimum eigenvalue suggests singularity, rank deficiency, or a nearly flat direction. Small rounding changes may matter more in that case.
4. Why is the Gershgorin lower bound included?
It provides a fast theoretical lower estimate for where eigenvalues can lie. It is useful for quick screening before relying on the final iterative solution.
5. How should I choose the tolerance?
Use a smaller tolerance for more precise diagonalization, but expect more iterations. For most moderate matrices, 1e-10 is a good default starting point.
6. Does this tool handle large matrices?
This page is designed for interactive matrices up to 6×6. That keeps the interface readable and the iterative computation quick in a browser-driven workflow.
7. What is the smallest eigenvector used for?
It identifies the direction associated with the minimum Rayleigh quotient. That can reveal weakest stiffness, lowest curvature, or the most sensitive mode.
8. Why might convergence fail within the iteration limit?
A very strict tolerance or a poorly scaled matrix can slow Jacobi rotations. Increasing the iteration cap or rescaling entries often helps.