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Beyond 2×2, there's no closed-form formula for a matrix's eigenvalues — they have to be found numerically. The QR algorithm is the standard method used in practice: applied to a matrix's Hessenberg form with shifts, it converges toward the eigenvalues on the diagonal (or in 2×2 blocks, for complex-conjugate pairs).
This calculator finds all eigenvalues — real or complex — of any square matrix up to 6×6, complementing the closed-form Eigenvalues (2×2) Calculator for larger matrices.
How it works
Enter a square matrix. The tool reduces it to Hessenberg form, then applies repeated shifted QR steps, deflating out eigenvalues as they converge. Any 2×2 block that resists further reduction represents a genuine complex-conjugate pair, which is solved exactly via the quadratic formula rather than iterated further.
- Click Calculate to see your results.
Examples
A matrix with a complex pair
For A = [[0,−1],[1,0]] (a 90° rotation), the eigenvalues are the complex-conjugate pair +i and −i — there are no real eigenvalues, since a 90° rotation has no direction it leaves unchanged.
Who should use it
- Analyzing the stability of a dynamical system (via the sign/magnitude of its eigenvalues).
- Finding eigenvalues of a non-symmetric matrix, where the Jacobi-based Symmetric Matrix Eigenvalue Calculator doesn't apply.
Industry applications
- Control theory and systems engineering
- Numerical linear algebra and scientific computing
Advantages
- Works for any square matrix up to 6×6, symmetric or not.
- Correctly handles complex-conjugate eigenvalue pairs, not just real eigenvalues.
Limitations
- Numerical (approximate) results, not exact symbolic ones, for matrices larger than 2×2.
Common mistakes to avoid
- Expecting an exact, closed-form answer the way you'd get from the quadratic formula for a 2×2 matrix — for larger matrices, numerical iteration is fundamentally required.
Best practices
- Check the trace displayed alongside the eigenvalues — it should equal the sum of all the eigenvalues (including complex ones, whose imaginary parts cancel in conjugate pairs) as a sanity check.
Tips
- If your matrix is symmetric, use the Symmetric Matrix Eigenvalue Calculator instead — the Jacobi method it uses is guaranteed to converge and also returns eigenvectors, which this general-purpose tool doesn't.