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Let the real QR shift be $\mu$ and put $r=\sqrt{1+{\mu }^2}$. A valid QR factorization, including the singular cases $\mu =\pm 1$, isHere $Q^{T}Q=I$, and multiplying verifies $QR=A-\mu I$. Using this orthogonality in the shifted update givesIt is diagonal exactly when ${\mu }^2-1=0$. Therefore all real shifts giving the eigenvalues in one iteration areThe resulting diagonal is $(-1,1)$ or $(1,-1)$, respectively. Both QR shifts are exact eigenvalues. Sign changes in the QR factorization conjugate $A_1$ by a diagonal sign matrix and do not alter its zero pattern. This is Exact-shift diagonalization of a two-state swap matrix.

A QR shift is a scalar ${\mu }_k$ subtracted before the QR factorization in a QR algorithm step and restored after reversing the factors:The final identity uses the orthogonality of $Q_k$, proving that the eigenvalues are preserved. Choosing a shift near an eigenvalue can accelerate block deflation in the QR algorithm. A common choice is the current bottom-right entry, but it can stall, as in Exact-shift diagonalization of a two-state swap matrix.