Hessenberg Reduction

A matrix A can be transformed using an orthogonal similarity transformation Q^TAQ. This operation changes the basis of the vector space without altering the matrix's eigenvalues, trace, or determinant. In this experiment, we compute a single Givens rotation to zero out the A₃₁ entry, which is sufficient to reach upper Hessenberg form for a 3×3 matrix. The resulting matrix is not necessarily upper triangular or in Schur form.

Match the product to the displayed controls

Open Hessenberg Reduction. Matrix A has nine integer sliders.

1. Set A₁₁ = 1, A₁₂ = 2, A₁₃ = 0, A₂₁ = 3, A₂₂ = 4, A₂₃ = 1, A₃₁ = 4, A₃₂ = 0, and A₃₃ = 2.
2. Observe the orthogonal matrix Q. Its entries are computed using c = 3/5 and s = 4/5.
3. Examine the intermediate Q^TA matrix. You will see that its (3,1) entry is exactly 0.
4. Check the final Hessenberg H = Q^TAQ. The (3,1) entry remains 0, while the trace (7) and determinant (4) match the original matrix A.

By adjusting A₃₁ to 0, you can see that Q becomes the identity matrix, and H equals A, meaning no reduction is needed.