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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 non-Hermitian. 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   Index
Susan Blackford 2000-11-20