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

Definition

; the definition can be generalized to rngs be a positive integer. Then any matrix-valued function

) is called a matrix pencil.

Polynomial matrix pencils

is polynomial). Then the polynomial matrix pencil (often simply a matrix pencil defined by

The degree zero matrix.

Linear matrix pencils

).).

Generalized eigenvalues of matrix pencils

is called a generalized eigenvalue (often simply eigenvalue is called its spectrum and is denoted by

).

are precisely the matrix eigenvalues is said to have one or more eigenvalues at infinity has one or more 0 eigenvalues.

A pencil is called regular; 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 is singular, or numerically unstable when it is ill-conditioned).

Pencils generated by commuting matrices

:

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

References

• Peter Lancaster & Qian Ye (1991) "Variational and numerical methods for symmetric matrix pencils", Bulletin of the Australian Mathematical Society 43: 1 to 17