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

Definition

Let K be a field (typically, ; the definition can be generalized to rngs), let \ell \ge 0 be a non-negative integer, let n > 0 be a positive integer, and let be n\times n matrices (i. e. for all ). Then the matrix pencil defined by is the matrix-valued function defined by The degree of the matrix pencil is defined as the largest integer such that A_k \ne 0 (the n \times n zero matrix over K).

Linear matrix pencils

A particular case is a linear matrix pencil (where B \ne 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).

Properties

A pencil is called regular if there is at least one value of \lambda such that ; otherwise it is called singular. We call eigenvalues of a matrix pencil all (complex) numbers \lambda for which ; in particular, the eigenvalues of the matrix pencil (A, I) are the matrix eigenvalues of 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. For the linear pencil (A, B), it is written as (not ). The linear pencil (A, B) is said to have one or more eigenvalues at infinity if 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 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: