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Since $A_k=Q_k{R}_k$,Thus every step is an orthogonal similarity, so it preserves eigenvalues and symmetry.

For a symmetric $r$-banded matrix, its subdiagonal entries can be eliminated by Givens rotations ordered along the band. Each rotation creates only the next local bulge; multiplication in reverse order chases that bulge out without creating entries beyond the original upper band. Symmetry supplies the corresponding lower band. This is symmetric bandwidth preservation under QR iteration, and proves that $A_{k+1}$ is again $r$-banded.