Regular matrix pencils and homogeneous generalized eigenvalues

Let $n\ge 1$, let $\mathbb{F}\in \{\mathbb{R},\mathbb{C}\}$, and let $A,B\in {\mathbb{F}}^{n\times n}$. The matrix pencil determined by $(A,B)$ is $A-\lambda B$. Its determinant polynomial is $f(\lambda )=\det (A-\lambda B)$, with real coefficients viewed as complex coefficients when $\mathbb{F}=\mathbb{R}$. The pencil is regular if $f$ is not the zero polynomial; otherwise it is singular.

For $\alpha ,\beta \in \mathbb{C}$ not both zero, put $p(\alpha ,\beta )=\det (\beta A-\alpha B)\mathrm{.}$ A homogeneous generalized eigenvalue pair is an equivalence class of such pairs satisfying $p(\alpha ,\beta )=0$, where $(\alpha ,\beta )$ and $(c\alpha ,c\beta )$ represent the same pair for every $c\in \mathbb{C}\setminus \{0\}$. If $\beta \mathrm{\ne }0$, the associated finite generalized eigenvalue is $\lambda =\alpha \mathrm{/}\beta$; if $\beta =0$, the pair is an infinite generalized eigenvalue. The pair $(0,0)$ is never an eigenvalue pair. Write $\sigma (A,B)$ for the set of finite ratios $\alpha \mathrm{/}\beta$ among these homogeneous eigenvalue pairs.

The QZ results on this page assume a regular pencil. They make no assertion about Kronecker structure or eigenvalues of a singular pencil.