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In linear algebra, if A 0 , A 1 , … , A ℓ {\displaystyle A_{0},A_{1},\dots ,A_{\ell }} are n × n {\displaystyle n\times n} complex matrices for some nonnegative integer ℓ {\displaystyle \ell } , and A ℓ ≠ 0 {\displaystyle A_{\ell }\neq 0} , then the matrix pencil of degree ℓ {\displaystyle \ell } is the matrix-valued function defined on the complex numbers L = ∑ i = 0 ℓ λ i A i . {\displaystyle L=\sum _{i=0}^{\ell }\lambda ^{i}A_{i}.}
A particular case is a linear matrix pencil A − λ B {\displaystyle A-\lambda B\,} with λ ∈ C , {\displaystyle \lambda \in \mathbb {C} {\text{ ,}}} where A {\displaystyle A} and B {\displaystyle B} are complex n × n {\displaystyle n\times n} matrices. We denote it briefly with the notation {\displaystyle }.
A pencil is called regular if there is at least one value of λ {\displaystyle \lambda } such that det ≠ 0 {\displaystyle \det\neq 0}. We call eigenvalues of a matrix pencil {\displaystyle } all complex numbers λ {\displaystyle \lambda } for which det = 0 {\displaystyle \det=0} . The set of the eigenvalues is called the spectrum of the pencil and is written σ {\displaystyle \sigma }.Moreover, the pencil is said to have one or more eigenvalues at infinity if B {\displaystyle B} has one or more 0 eigenvalues.