Matrix pencil

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

Definition

Let $K$ be a field (typically, $K\in \{\mathbb {R} ,\mathbb {C} \}$ ; the definition can be generalized to 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 $B\neq 0$ ).^[1] 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 $B$ ).

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

Pencils generated by commuting matrices

If $AB=BA$ , then the pencil generated by $A$ and $B$ :^[2]

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.

Definition

Polynomial matrix pencils

Linear matrix pencils

Generalized eigenvalues of matrix pencils

Applications

Pencils generated by commuting matrices

See also

Notes

References

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