= Eigenvalue at infinity
{title2=$\lambda=\infty$}
= Eigenvalues at infinity
{synonym}
For a regular <matrix pencil> $A-\lambda B$, an <eigenvalue at infinity> is represented by a homogeneous pair $(\alpha,\beta)$ with $\beta=0$ and $\alpha\ne0$. The <eigenvector> relation $\beta Av=\alpha Bv$ then requires $Bv=0$ with $v\ne0$.
For $A=I$ and $B=\operatorname{diag}(1,0)$, 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 $n-\deg\det(A-\lambda B)$. 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>.<Eigenvalues and Schur Forms in the LAPACK Users' Guide>[Generalized nonsymmetric eigenproblems: homogeneous pairs and regular versus singular pencils]{ref}
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