Past exam of the mathematics course of the University of Cambridge / 2025 / ib / Paper 1 / 17A / d

For symmetric $A$, apply at stage $k$ the same reflection on both sides, choosing it to zero entries below the first subdiagonal in column $k$. Orthogonal similarity preserves symmetry, so the corresponding row entries vanish too and previous zeros remain. After finitely many stages $QA{Q}^T$ is symmetric tridiagonal. Constructing each reflector uses only arithmetic and one square root.