= Real Schur form
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 $\begin{pmatrix}0&-1\\1&0\end{pmatrix}$ 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>.<Eigenvalues and Schur Forms in the LAPACK Users' Guide>[Ordinary Schur factorization and real diagonal blocks]{ref}
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