Matrix pencil explained
, usually the real or complex numbers.
Definition
Let
be a field (typically,
; the definition can be generalized to
rngs), let
be a non-negative integer, let
be a positive integer, and let
be
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
(the
zero matrix over
).
Linear matrix pencils
A particular case is a linear matrix pencil
(where
). We denote it briefly with the notation
, and note that using the more general notation,
and
(not
).
Properties
A pencil is called regular if there is at least one value of
such that
; otherwise it is called
singular. We call
eigenvalues of a matrix pencil all (complex) numbers
for which
; 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
, it is written as
(not
).
The linear pencil
is said to have one or more eigenvalues
at infinity if
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
(which is impossible when
is singular, or numerically unstable when it is
ill-conditioned).
Pencils generated by commuting matrices
If
, then the pencil generated by
and
:
- consists only of matrices similar to a diagonal matrix, or
- has no matrices in it similar to a diagonal matrix, or
- has exactly one matrix in it similar to a diagonal matrix.
See also
References
