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Hessenberg form is almost triangular: every entry below the first subdiagonal is zero. Reducing a matrix to this form via a similarity transform (H = QᵀAQ) doesn't change its eigenvalues, and makes subsequent eigenvalue algorithms — like the QR algorithm — converge dramatically faster.
This calculator reduces any square matrix up to 6×6 to upper Hessenberg form, showing both H and the orthogonal transform Q used to get there.
How it works
Enter a square matrix. The tool applies a sequence of Householder reflections on both sides of the matrix (a similarity transform), each one zeroing out one column below its first subdiagonal entry, until the whole matrix is in upper Hessenberg form.
- Click Calculate to see your results.
Examples
A 4×4 example
For a symmetric 4×4 matrix, Hessenberg reduction produces a tridiagonal matrix (zero everywhere except the diagonal and the two adjacent diagonals) with the same eigenvalues as the original.
Who should use it
- Understanding the preprocessing step used by numerical eigenvalue algorithms.
- Verifying a similarity transform preserves a matrix's trace and eigenvalue structure.
Industry applications
- Numerical linear algebra and scientific computing
Advantages
- Shows a key intermediate step used inside real eigenvalue algorithms.
- Preserves eigenvalues exactly, unlike an arbitrary row-reduction.
Limitations
- Not a final answer on its own — it's a preprocessing step toward eigenvalues, not the eigenvalues themselves.
Common mistakes to avoid
- Expecting Hessenberg form itself to reveal the eigenvalues directly — it doesn't, until further iteration (like the QR algorithm) is applied on top of it.
Best practices
- Use this as a diagnostic or educational tool to see the intermediate step that eigenvalue algorithms perform internally — the Eigenvalues Calculator (General Matrix) and Schur Decomposition Calculator do this automatically as part of their own computation.
Tips
- For a symmetric input matrix, Hessenberg form is automatically tridiagonal (zero everywhere except three central diagonals) — a useful sanity check on your result.