In linear algebra, a matrix pencil is a matrix-valued function defined on a field K {\displaystyle K} , usually the real or complex numbers.
Definition Let K {\displaystyle K} be a field (typically, K ∈ { R , C } {\displaystyle K\in \{\mathbb {R} ,\mathbb {C} \}} ; the definition can be generalized to rngs, i.e. non-unital rings), and let n > 0 {\displaystyle n>0} be a positive integer. Then any matrix-valued function
P : K → M a t ( K , n × n ) {\displaystyle P\colon K\to \mathrm {Mat} (K,n\times n)}
(where M a t ( K , n × n ) {\displaystyle \mathrm {Mat} (K,n\times n)} denotes the K {\displaystyle K} -algebra of n × n {\displaystyle n\times n} matrices over K {\displaystyle K} ) is called a matrix pencil.
Polynomial matrix pencils An important special case arises when P {\displaystyle P} is polynomial: let ℓ ≥ 0 {\displaystyle \ell \geq 0} be a non-negative integer, and let A 0 , A 1 , … , A ℓ {\displaystyle A_{0},A_{1},\dots ,A_{\ell }} be n × n {\displaystyle n\times n} matrices (i. e. A i ∈ M a t ( K , n × n ) {\displaystyle A_{i}\in \mathrm {Mat} (K,n\times n)} for all i = 0 , … , ℓ {\displaystyle i=0,\dots ,\ell } ). Then the polynomial matrix pencil (often simply a matrix pencil) defined by A 0 , … , A ℓ {\displaystyle A_{0},\dots ,A_{\ell }} is the matrix-valued function L : K → M a t ( K , n × n ) {\displaystyle L\colon K\to \mathrm {Mat} (K,n\times n)} defined by
L ( λ ) = ∑ i = 0 ℓ λ i A i . {\displaystyle L(\lambda )=\sum _{i=0}^{\ell }\lambda ^{i}A_{i}.}
The degree of this matrix pencil is defined as the largest integer 0 ≤ k ≤ ℓ {\displaystyle 0\leq k\leq \ell } such that A k ≠ 0 {\displaystyle A_{k}\neq 0} , the n × n {\displaystyle n\times n} zero matrix over K {\displaystyle K} .
Linear matrix pencils A particular case is a linear matrix pencil L ( λ ) = A − λ B {\displaystyle L(\lambda )=A-\lambda B} (where B ≠ 0 {\displaystyle B\neq 0} ). We denote it briefly with the notation ( A , B ) {\displaystyle (A,B)} , and note that using the more general notation, A 0 = A {\displaystyle A_{0}=A} and A 1 = − B {\displaystyle A_{1}=-B} (not B {\displaystyle B} ).
Generalized eigenvalues of matrix pencils For a matrix pencil P {\displaystyle P} , any k ∈ K {\displaystyle k\in K} such that det P ( k ) = 0 K {\displaystyle \det P(k)=0_{K}} is called a generalized eigenvalue (often simply eigenvalue) of P {\displaystyle P} , and the set of generalized eigenvalues of P {\displaystyle P} is called its spectrum and is denoted by
σ ( P ) = { k ∈ K : det P ( k ) = 0 K } . {\displaystyle \sigma (P)=\{k\in K:\det P(k)=0_{K}\}.}
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