 Solution

The initial bottom-right entry is zero, so the chosen QR shift is ${\mu }_0=0$. One valid QR factorization is $Q_0=A$, $R_0=I$, using explicitly that $A^{T}A=I$. The update isIt repeats forever and never approaches a diagonal matrix. With the standard positive-diagonal convention for $R$, this factorization is unique, so the Rayleigh-shift iteration is stationary and fails to diagonalize $A$.

The failure is independent of harmless sign conventions. The columns of $A$ are already orthonormal, so another QR factorization can only replace $Q_0$ by $AD$ and $R_0$ by $D$, where $D$ is diagonal with entries $\pm 1$. Then $A_1=DAD$ still has zero diagonal and off-diagonal magnitudes one. Subsequent bottom-right shifts remain zero and the off-diagonal magnitude never decreases. The eigenvalues are $+1,-1$, with equal absolute values, explaining the lack of separation in the unshifted QR algorithm.