Matrix pencil

In linear algebra, a matrix pencil is a matrix-valued polynomial function defined on a field

, usually the real or complex numbers.

be a field (typically,

; the definition can be generalized to rngs), let

be a non-negative integer, let

be a positive integer, and let

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

A particular case is a linear matrix pencil

[1] We denote it briefly with the notation

, and note that using the more general notation,

A pencil is called regular if there is at least one value of

; otherwise it is called singular.

We call eigenvalues of a matrix pencil all (complex) numbers

; in particular, the eigenvalues of the matrix pencil

are the matrix eigenvalues of

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

The linear pencil

is said to have one or more eigenvalues at infinity if

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

is singular, or numerically unstable when it is ill-conditioned).

, then the pencil generated by

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