Encyclopedia

Matrix pencil

Concept in linear algebra

Wikipedia contributorsAdapted for Fathomly · View authors

2 min read
Source updated August 24, 2026

In linear algebra, a matrix pencil is a matrix-valued function defined on a field $K$ , usually the real or complex numbers.

01Definition

Let $K$ be a field (typically, $K\in \{\mathbb {R} ,\mathbb {C} \}$ ; the definition can be generalized to rngs, i.e. non-unital rings), and let $n>0$ be a positive integer. Then any matrix-valued function

$P\colon K\to \mathrm {Mat} (K,n\times n)$

(where $\mathrm {Mat} (K,n\times n)$ denotes the $K$ -algebra of $n\times n$ matrices over $K$ ) is called a matrix pencil.

Polynomial matrix pencils

An important special case arises when $P$ is polynomial: let $\ell \geq 0$ be a non-negative integer, and let $A_{0},A_{1},\dots ,A_{\ell }$ be $n\times n$ matrices (i. e. $A_{i}\in \mathrm {Mat} (K,n\times n)$ for all $i=0,\dots ,\ell$ ). Then the polynomial matrix pencil (often simply a matrix pencil) defined by $A_{0},\dots ,A_{\ell }$ is the matrix-valued function $L\colon K\to \mathrm {Mat} (K,n\times n)$ defined by

$L(\lambda )=\sum _{i=0}^{\ell }\lambda ^{i}A_{i}.$

The degree of this matrix pencil is defined as the largest integer $0\leq k\leq \ell$ such that $A_{k}\neq 0$ , the $n\times n$ zero matrix over $K$ .

Linear matrix pencils

A particular case is a linear matrix pencil $L(\lambda )=A-\lambda B$ (where $B\neq 0$ ). We denote it briefly with the notation $(A,B)$ , and note that using the more general notation, $A_{0}=A$ and $A_{1}=-B$ (not $B$ ).

02Generalized eigenvalues of matrix pencils

For a matrix pencil $P$ , any $k\in K$ such that $\det P(k)=0_{K}$ is called a generalized eigenvalue (often simply eigenvalue) of $P$ , and the set of generalized eigenvalues of $P$ is called its spectrum and is denoted by

$\sigma (P)=\{k\in K:\det P(k)=0_{K}\}.$

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

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

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

03Applications

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 $Ax=\lambda Bx$ without inverting the matrix $B$ (which is impossible when $B$ is singular, or numerically unstable when it is ill-conditioned).

04Pencils generated by commuting matrices

If $AB=BA$ , then the pencil generated by $A$ and $B$ :

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

Watch videos about Matrix pencilExplainers and documentaries on YouTube (opens in a new tab)

Sources and credits

This article is adapted from the Wikipedia article “Matrix pencil”, written by its contributors and licensed under CC BY-SA 4.0. Fathomly has changed the layout, removed citation markers, navigation and maintenance notices, and adjusted punctuation. This adapted version is shared under the same license. For references, see the original article.

Fathomly is not affiliated with or endorsed by the Wikimedia Foundation. Spotted a problem? Tell us.

Matrix pencil

01Definition

Polynomial matrix pencils

Linear matrix pencils

02Generalized eigenvalues of matrix pencils

03Applications

04Pencils generated by commuting matrices

Sources and credits

Related topics

Eigendecomposition of a matrix

Generalized pencil-of-function method

Nonlinear eigenproblem