Matrix pencil

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

Definition

Let ${\displaystyle K}$ be a field (typically, ${\displaystyle K\in \{\mathbb {R} ,\mathbb {C} \}}$; the definition can be generalized to rngs), let ${\displaystyle \ell \geq 0}$ be a non-negative integer, let ${\displaystyle n>0}$ be a positive integer, and let ${\displaystyle A_{0},A_{1},\dots ,A_{\ell }}$ be ${\displaystyle n\times n}$ matrices (i. e. ${\displaystyle A_{i}\in \mathrm {Mat} (K,n\times n)}$ for all ${\displaystyle i=0,\dots ,\ell }$). Then the matrix pencil defined by ${\displaystyle A_{0},\dots ,A_{\ell }}$ is the matrix-valued function ${\displaystyle L\colon K\to \mathrm {Mat} (K,n\times n)}$ defined by

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

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

Linear matrix pencils

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

Properties

A pencil is called regular if there is at least one value of ${\displaystyle \lambda }$ such that ${\displaystyle \det(L(\lambda ))\neq 0}$; otherwise it is called singular. We call eigenvalues of a matrix pencil all (complex) numbers ${\displaystyle \lambda }$ for which ${\displaystyle \det(L(\lambda ))=0}$; in particular, the eigenvalues of the matrix pencil ${\displaystyle (A,I)}$ are the matrix eigenvalues of ${\displaystyle A}$. For linear pencils in particular, the eigenvalues of the pencil are also called generalized eigenvalues.

The set of the eigenvalues of a pencil is called the spectrum of the pencil, and is written ${\displaystyle \sigma (A_{0},\dots ,A_{\ell })}$. For the linear pencil ${\displaystyle (A,B)}$, it is written as ${\displaystyle \sigma (A,B)}$ (not ${\displaystyle \sigma (A,-B)}$).

The linear pencil ${\displaystyle (A,B)}$ is said to have one or more eigenvalues at infinity if ${\displaystyle B}$ has one or more 0 eigenvalues.

Applications

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

Pencils generated by commuting matrices

If ${\displaystyle AB=BA}$, then the pencil generated by ${\displaystyle A}$ and ${\displaystyle B}$:[2]

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

Notes

1. Golub & Van Loan (1996, p. 375)
2. Marcus & Minc (1969, p. 79)

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