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}\}.}

For a polynomial matrix pencil, we write σ ( A 0 , … , A ℓ ) {\displaystyle \sigma (A_{0},\dots ,A_{\ell })}; for the linear pencil ( A , B ) {\displaystyle (A,B)}, we write as σ ( A , B ) {\displaystyle \sigma (A,B)} (not σ ( A , − B ) {\displaystyle \sigma (A,-B)}).

The generalized eigenvalues of the linear matrix pencil ( A , I ) {\displaystyle (A,I)} are precisely the matrix eigenvalues of A {\displaystyle A}. The general linear pencil ( A , B ) {\displaystyle (A,B)} is said to have one or more eigenvalues at infinity if B {\displaystyle B} has one or more 0 eigenvalues.

A pencil is called regular if there is at least one k ∈ K {\displaystyle k\in K} such that det P ( k ) ≠ 0 K {\displaystyle \det P(k)\neq 0_{K}}, i. e. if λ ( P ) ≠ K {\displaystyle \lambda (P)\neq K}; otherwise it is called singular.

Applications

Matrix pencils play an important role in numerical linear algebra. The problem of finding the generalized eigenvalues of a pencil is called the generalized eigenvalue problem. The most popular algorithm for this task is the QZ algorithm, an implicit version of the QR algorithm for solving the eigenvalue problem A x = λ B x {\displaystyle Ax=\lambda Bx} without inverting the matrix B {\displaystyle B} (which is impossible when B {\displaystyle B} is singular, or numerically unstable when it is ill-conditioned).

Pencils generated by commuting matrices

If A B = B A {\displaystyle AB=BA}, then the pencil generated by A {\displaystyle A} and B {\displaystyle B}:

  1. consists only of matrices similar to a diagonal matrix, or
  2. has no matrices in it similar to a diagonal matrix, or
  3. has exactly one matrix in it similar to a diagonal matrix.

See also

Notes