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Transformation to Linear Form
It is easy to see that the QEP in (9.1) is equivalent to
the following generalized
``linear'' eigenvalue problem:^{}

(248) 
where

(249) 
and

(250) 
The generalized eigenvalue problem (9.4)
is commonly called a linearization of the QEP (9.1).
It can be shown that for any matrices and of the above forms,
the right and left eigenvectors and have the structures
described in (9.6).
Note that from the factorization

(251) 
we can conclude that the pencil is equivalent^{} to the matrix

(252) 
and
This means that the eigenvalues of the original QEP (9.1)
coincide with the eigenvalues of the generalized eigenvalue
problem (9.4). Furthermore, we have that
 is regular if and only if is regular;
 if (hence ) is nonsingular, then is regular;
 if (hence ) is nonsingular, then is regular.
For the theory on regular pencils , see, for instance,
[425, Chap. VI].
We will assume that at least is nonsingular throughout this section.
A disadvantage of the above reduction to linear form is that if
the matrices , , and are all Hermitian, then this is not reflected in
the reduced form (9.5), where is
nonHermitian. This can be repaired as follows.
In fact, the matrix pair in (9.4) can be
chosen in a more general form
where can be any arbitrary nonsingular matrix.
Note that now the matrix pencil is equivalent to the
matrix polynomial (9.8) if and only if is nonsingular,
and because of (9.7),
For example, if the matrices , , and are all symmetric, as
in the special case (9.2), and is nonsingular, then we may choose
, which leads to the following symmetric generalized ``linear''
eigenvalue problem

(253) 
where

(254) 
and

(255) 
Both and are symmetric, but may be indefinite.
Next: Spectral Transformations for QEP
Up: Quadratic Eigenvalue Problems Z. Bai,
Previous: Introduction
Contents
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Susan Blackford
20001120