 Eigenvalue at infinity

For a regular matrix pencil $A-\lambda B$, an eigenvalue at infinity is represented by a homogeneous pair $(\alpha ,\beta )$ with $\beta =0$ and $\alpha \ne 0$. The eigenvector relation $\beta Av=\alpha Bv$ then requires $Bv=0$ with $v\ne 0$.

For $A=I$ and $B=\mathrm{diag}(1,0)$, the first coordinate has finite eigenvalue one and the second has an infinite eigenvalue. More generally, if $A,B$ are $n\times n$ and the pencil is regular, its infinite algebraic multiplicity is $n-\mathrm{deg}\det (A-\lambda B)$. A singular pencil, whose determinant is identically zero, requires a different structural analysis and cannot be treated as an ordinary list of finite and infinite eigenvalues.^[ref]