Let $Q \in \mathbb{C}^{n\times n}$ be a unitary matrix that can be exactly stored in floating point arithmetic. Suppose we want to solve the following linear system: \begin{equation} Qx=b \end{equation} Given that $Q$ is unitary, the simplest solution is $x=Q^{*}b$. In floating point arithmetic $Qfl(Q^{*}b)$ is not exactly $b$. It turns out that exist $||\delta Q|| = O(\epsilon_{machine})$ such that \begin{equation} (Q + \delta Q)fl(Q^{*}b)=b \end{equation}
This result is stated without proof on the book Numerical Linear Algebra (Trefthen and Bau) in chapter 16.
I tried to formulate this as the problem of finding $A \in \mathbb{C}^{n\times n}$, such that \begin{equation} Afl(Q^{* }b)=b \end{equation} But I was not able to show that $A=Q + \delta Q$ for some $\delta Q$ with $||\delta Q||=O(\epsilon_{machine})$ is the solution for that problem.