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 $\left(\begin{array}{cc}0& -1\\ 1& 0\end{array}\right)$ 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]