This paper claimed that (in section 1) the QR iteration preserves the lower bandwidth of $A$, where $A \in \mathbb{C}^{n \times n}$ is the decomposed matrix, but I cannot prove it.
I formulate the problem as follows.
Prove: Suppose $A$ has the lower bandwidth $p$, if we perform the (shifted) QR iteration \begin{equation} A - \mu I = QR, \quad \bar{A} = RQ+\mu I, \end{equation} then $\bar{A}$ also has lower bandwidth $p$.
Any help will be appreciated.