Matrix pencil
Encyclopedia
If 
are 
complex matrices for some nonnegative integer 
, and 
(the zero matrix), then the matrix pencil
of degree
is the matrix-valued function defined on the complex numbers
A particular case is a linear matrix pencil:
with
where
and 
are complex (or real) 
matrices
. We denote it briefly with the notation
A pencil is called regular if there is at least one value of
such that 
. We call eigenvalues of a matrix pencil 
all complex numbers 
for which 
(see eigenvalue for comparison). The set of the eigenvalues is called the spectrum of the pencil and is written 
.
Moreover, the pencil is said to have one or more eigenvalues at infinity if
has one or more 0 eigenvalues.
. 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 associated eigenvalue problem
without forming explicitly the matrix 
(which could be impossible or ill-conditioned if 
is singular or near-singular)
, then the pencil generated by 
and 
(Marcus & Minc, 1969, p. 79):
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.
are complex matrices for some nonnegative integer , and (the zero matrix), then the matrix pencilPencil (mathematics)
A pencil in projective geometry is a family of geometric objects with a common property, for example the set of lines that pass through a given point in a projective plane....
of degree
is the matrix-valued function defined on the complex numbersA particular case is a linear matrix pencil:
with
where
and are complex (or real) matricesMatrix (mathematics)
In mathematics, a matrix is a rectangular array of numbers, symbols, or expressions. The individual items in a matrix are called its elements or entries. An example of a matrix with six elements isMatrices of the same size can be added or subtracted element by element...
. We denote it briefly with the notation
A pencil is called regular if there is at least one value of
such that . We call eigenvalues of a matrix pencil all complex numbers for which (see eigenvalue for comparison). The set of the eigenvalues is called the spectrum of the pencil and is written .Moreover, the 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 algebraNumerical linear algebra
Numerical linear algebra is the study of algorithms for performing linear algebra computations, most notably matrix operations, on computers. It is often a fundamental part of engineering and computational science problems, such as image and signal processing, Telecommunication, computational...
. 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
QR algorithm
In numerical linear algebra, the QR algorithm is an eigenvalue algorithm: that is, a procedure to calculate the eigenvalues and eigenvectors of a matrix. The QR transformation was developed in the late 1950s by John G.F. Francis and by Vera N. Kublanovskaya , working independently...
to solve the associated eigenvalue problem
without forming explicitly the matrix (which could be impossible or ill-conditioned if is singular or near-singular)Pencil generated by commuting matrices
If, then the pencil generated by and (Marcus & Minc, 1969, p. 79):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.