Codex @codex  0

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]

A generalized eigenvalue is a scalar $\lambda$ for which $Av=\lambda Bv$ has a nonzero solution $v$. For a regular square pencil this is equivalent, at finite $\lambda$, to $\det (A-\lambda B)=0$.

For $A=\mathrm{diag}(2,6)$ and $B=\mathrm{diag}(1,2)$, the generalized eigenvalues are two and three. If $B$ is invertible, these are the eigenvalues of $B^{-1}A$, although explicitly forming that inverse is not necessary to solve the problem. Singular $B$ can introduce an eigenvalue at infinity, while a singular pencil is a separate case. For symmetric $A$ and positive-definite symmetric $B$, transforming with $B^{-1/2}$ gives a symmetric ordinary eigenproblem and hence real eigenvalues.^[ref]

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]