Symmetric Eigenvalues Calculator for Real Matrices

Calculator

Matrix Size

Decimal Places

Sort Order

Tolerance

Max Iterations

Symmetry Handling

Matrix Entries

Edit any cell. Mirror mode copies values across the diagonal while typing.

The calculator uses a Jacobi rotation method. It is designed for real symmetric matrices, where eigenvalues are real and eigenvectors can be chosen orthonormal.

Example Data Table

Example	Symmetric Matrix	Ordered Eigenvalues	Classification	Trace
Sample A	[4 1 2; 1 3 0; 2 0 5]	6.669079, 3.476024, 1.854897	Positive definite	12
Sample B	[2 0 0; 0 5 1; 0 1 4]	5.618034, 3.381966, 2.000000	Positive definite	11

Formula Used

For a real symmetric matrix A, the calculator finds eigenvalues and eigenvectors from the characteristic relation and orthogonal diagonalization.

det(A - λI) = 0

A = QΛQᵀ

Λ = diag(λ₁, λ₂, ..., λₙ)

trace(A) = Σλᵢ

det(A) = Πλᵢ

The computational routine uses Jacobi rotations to iteratively eliminate off-diagonal terms. For symmetric matrices, this process converges to a diagonal matrix of eigenvalues, while the accumulated rotation matrix contains orthonormal eigenvectors.

How to Use This Calculator

Select a matrix size from 2 × 2 up to 6 × 6.
Enter matrix values. Keep entries symmetric around the main diagonal.
Choose decimal precision, tolerance, iteration limit, and sort order.
Use mirror mode for faster entry or strict mode for validation.
Press Calculate Eigenvalues to generate results above the form.
Review eigenvalues, eigenvectors, trace, determinant, condition estimate, and classification.
Use CSV or PDF export buttons for reporting or record keeping.

Frequently Asked Questions

1. What matrix types work with this calculator?

It is built for real symmetric matrices only. Symmetric means the value in row i, column j equals the value in row j, column i. This condition guarantees real eigenvalues and orthogonal eigenvectors.

2. Why are symmetric matrices easier to analyze?

They have strong numerical properties. Their eigenvalues are always real, and their eigenvectors can be chosen orthonormal. That makes decomposition stable, interpretable, and useful for optimization, geometry, and quadratic forms.

3. What does the classification result mean?

The classification comes from the eigenvalue signs. All positive means positive definite. All negative means negative definite. Mixed signs mean indefinite. Zeros can indicate semidefinite behavior or singularity.

4. Why does trace match the sum of eigenvalues?

This is a standard matrix invariant. For any square matrix, the trace equals the sum of all eigenvalues, counting multiplicity. The calculator shows both values so you can verify the result quickly.

5. Why does determinant match the product of eigenvalues?

This is another core invariant. For square matrices, the determinant equals the product of the eigenvalues. A near-zero determinant usually signals a nearly singular matrix and a weak smallest eigenvalue.

6. What is the condition estimate for?

It compares the largest and smallest absolute eigenvalues. A very large value suggests sensitivity, weak numerical stability, or a near-singular matrix. Infinite or undefined results usually mean one eigenvalue is extremely close to zero.

7. What are reconstruction and orthogonality errors?

Reconstruction error checks how closely QΛQᵀ reproduces the original matrix. Orthogonality error checks whether the eigenvectors remain close to an orthonormal basis. Smaller values indicate stronger numerical consistency.

8. Can I use this for principal component analysis ideas?

Yes. Symmetric eigenvalue problems appear in covariance matrices, inertia tensors, vibration models, and quadratic optimization. The ordered eigenvalues help rank dominant directions, while eigenvectors reveal those directions explicitly.