Matrix pencil


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

Definition

Let be a field, and let be a positive integer. Then any matrix-valued function
is called a '''matrix pencil.'''

Polynomial matrix pencils

An important special case arises when is polynomial: let be a non-negative integer, and let be matrices. Then the polynomial matrix pencil defined by is the matrix-valued function defined by
The degree of this matrix pencil is defined as the largest integer such that, the zero matrix over.

Linear matrix pencils

A particular case is a linear matrix pencil . We denote it briefly with the notation, and note that using the more general notation, and .

Generalized eigenvalues of matrix pencils

For a matrix pencil, any such that is called a generalized eigenvalue of, and the set of generalized eigenvalues of is called its spectrum and is denoted by
For a polynomial matrix pencil, we write ; for the linear pencil, we write as .
The generalized eigenvalues of the linear matrix pencil are precisely the matrix eigenvalues of. The general linear pencil is said to have one or more eigenvalues at infinity if has one or more 0 eigenvalues.
A pencil is called regular if there is at least one such that, i. e. if ; 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 without inverting the matrix .

Pencils generated by commuting matrices

If, then the pencil generated by and :
  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.