Matrix pencil

In linear algebra, if ${\displaystyle A_{0},A_{1},\dots ,A_{\ell }}$ are ${\displaystyle n\times n}$ complex matrices for some nonnegative integer ${\displaystyle \ell }$, and ${\displaystyle A_{\ell }\neq 0}$ (the zero matrix), then the matrix pencil of degree ${\displaystyle \ell }$ is the matrix-valued function defined on the complex numbers ${\displaystyle L(\lambda )=\sum _{i=0}^{\ell }\lambda ^{i}A_{i}.}$

A particular case is a linear matrix pencil ${\displaystyle A-\lambda B}$ with ${\displaystyle \lambda \in \mathbb {C} }$ (or ${\displaystyle \mathbb {R} }$) where ${\displaystyle A}$ and ${\displaystyle B}$ are complex (or real) ${\displaystyle n\times n}$ matrices.^[1] We denote it briefly with the notation ${\displaystyle (A,B)}$.

A pencil is called regular if there is at least one value of ${\displaystyle \lambda }$ such that ${\displaystyle \det(A-\lambda B)\neq 0}$. We call eigenvalues of a matrix pencil ${\displaystyle (A,B)}$ all complex numbers ${\displaystyle \lambda }$ for which ${\displaystyle \det(A-\lambda B)=0}$; in particular, the eigenvalues of the matrix pencil ${\displaystyle (A,I)}$ are the matrix eigenvalues of ${\displaystyle A}$. The set of the eigenvalues is called the spectrum of the pencil and is written ${\displaystyle \sigma (A,B)}$. Moreover, the pencil is said to have one or more eigenvalues at infinity if ${\displaystyle B}$ has one or more 0 eigenvalues.