Real and complex Schur forms

Let $\mathbb{F}$ be $\mathbb{R}$ or $\mathbb{C}$, let $n\ge 0$ and $A\in {\mathbb{F}}^{n\times n}$, and let $Q^{\ast }$ denote transpose over $\mathbb{R}$ and conjugate transpose over $\mathbb{C}$. A complex Schur pair for $A$ is a pair $(Q,T)$ with $Q^{\ast }Q=I$ and $A=QT{Q}^{\ast }$, where $T$ is upper triangular.

For real $A$, a real Schur pair is a pair $(Q,T)$ with $Q^{T}Q=I$ and $A=QT{Q}^T$, where $T$ is block upper triangular and every diagonal block is either a real $1\times 1$ block or a real $2\times 2$ block with negative characteristic-polynomial discriminant. Such a $2\times 2$ block has one nonreal conjugate eigenvalue pair. The definition does not assert that a finite shift strategy reaches such a pair.

For $B=\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)\in {\mathbb{R}}^{2\times 2}$, its characteristic-polynomial discriminant means the exact real number $\mathrm{\Delta }(B)=(a+d{)}^2-4(ad-bc)$. “Negative discriminant” means the exact comparison $\mathrm{\Delta }(B)<0$.