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 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 $B$).

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 $B$:

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

• Generalized eigenvalue problem
• Generalized pencil-of-function method
• Nonlinear eigenproblem
• Quadratic eigenvalue problem
• Generalized Rayleigh quotient

References

• Golub, Gene H.; Van Loan, Charles F. (1996), Matrix Computations (3rd ed.), Baltimore: Johns Hopkins University Press, ISBN 0-8018-5414-8
• Marcus & Minc (1969), A survey of matrix theory and matrix inequalities, Courier Dover Publications
• Peter Lancaster & Qian Ye (1991) "Variational and numerical methods for symmetric matrix pencils", Bulletin of the Australian Mathematical Society 43: 1 to 17