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The real Schur form of a real square matrix $A$ is $Q^{\top }AQ=T$, where $Q$ is orthogonal and $T$ is upper quasi-triangular. Its diagonal consists of one-by-one blocks for real eigenvalues and two-by-two blocks for complex-conjugate pairs.

The rotation generator $\left(\begin{array}{cc}0& -1\\ 1& 0\end{array}\right)$ already forms one two-by-two block with eigenvalues $\pm i$. No real upper-triangular matrix can have those eigenvalues, because its diagonal entries would all be real. Real Schur form therefore permits real arithmetic without pretending every eigenvalue is real. Its orthogonal transformations are also preferable numerically to using a potentially ill-conditioned eigenvector matrix.^[ref]

The unshifted QR algorithm starts at $A_0=A$, factors $A_k=Q_k{R}_k$, and formsfor real data with orthogonal $Q_k$. Each step is an orthogonal similarity, so it preserves eigenvalues. It is the zero-shift case of the QR algorithm; complex data use a unitary factor and conjugate transpose.

Convergence needs hypotheses. For a real symmetric matrix with suitably separated eigenvalue moduli and nondegenerate starting projections, the simultaneous iteration interpretation of the QR algorithm explains convergence toward diagonal form. Equal moduli can prevent that behavior: for $A=\left(\begin{array}{cc}0& 1\\ 1& 0\end{array}\right)$, the valid factorization $Q=A$, $R=I$ leaves every iterate equal to $A$. Its eigenvalues $1,-1$ remain coupled.

For nonsymmetric matrices, the useful target is generally a triangular form or a real Schur form, rather than a diagonal matrix. Shifts and block deflation in the QR algorithm improve practical convergence. The similarity identity by itself is not a convergence theorem.