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Symmetric matrices are a special, well-behaved case for eigenvalues: every eigenvalue is guaranteed to be real, and there's always a full set of mutually perpendicular eigenvectors. The Jacobi eigenvalue algorithm exploits this by repeatedly rotating away the largest off-diagonal entry until the matrix becomes (numerically) diagonal — a method that always converges, unlike general eigenvalue algorithms.
This calculator finds all eigenvalues and eigenvectors of a symmetric matrix up to 6×6.
How it works
Enter a symmetric matrix (a_ij must equal a_ji). The tool repeatedly applies a rotation that zeroes out the current largest off-diagonal entry, without changing the matrix's eigenvalues, until every off-diagonal entry is negligible — at which point the diagonal holds the eigenvalues and the accumulated rotations form the eigenvectors.
- Click Calculate to see your results.
Examples
A 2×2 symmetric example
For A = [[2,1],[1,2]], the Jacobi method finds eigenvalues 3 and 1, with eigenvectors (1,1)/√2 and (1,−1)/√2 respectively.
Who should use it
- Finding eigenvalues of covariance matrices (used in statistics and PCA).
- Diagonalizing symmetric matrices in physics and engineering (like moment of inertia tensors).
Industry applications
- Statistics and data science (principal component analysis)
- Numerical linear algebra and scientific computing
Advantages
- Always converges, unlike general (non-symmetric) eigenvalue algorithms.
- Returns both eigenvalues and a full orthonormal eigenvector basis in one step.
Limitations
- Only applicable to symmetric matrices — not a general-purpose eigenvalue solver.
Common mistakes to avoid
- Feeding in a non-symmetric matrix and expecting valid results — the calculator will detect this and ask you to use the general eigenvalue calculator instead.
- Forgetting that eigenvector signs aren't unique — a valid eigenvector times −1 is still a valid eigenvector for the same eigenvalue.
Best practices
- Verify your result by checking that A times an eigenvector equals that eigenvector's eigenvalue times itself (Av = λv).
Tips
- For a full spectral decomposition (A = PDP⁻¹ shown explicitly, with P and D as separate matrices), use the Eigendecomposition Calculator, which uses the same underlying method.