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The unshifted QR algorithm computes the eigenvalues of $A$. For a real symmetric matrix it drives $A_k$ toward a diagonal or block-diagonal matrix; the accumulated orthogonal factors simultaneously approximate its eigenvectors.

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.

The QR algorithm computes eigenvalues through repeated QR decompositions and similarity updates. Shifted and unshifted forms have different convergence rates; the unshifted QR algorithm uses a zero shift at every step.