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## Some Properties of Symmetric Indefinite Matrix Pairs

The symmetry of and is purely an algebraic property and is not sufficient to ensure any of the special mathematical properties enjoyed by a definite matrix pencil, such as those discussed in §2.3. In fact, it can be shown that any real square matrix may be written as or , where and are suitable symmetric matrices; for example, see [353].

The eigenvalues of a definite matrix pencil are all real, but an indefinite pencil may have complex eigenvalues. For example, when

the eigenvalues of are the complex conjugate pair . A further distinction from the definite case is that an indefinite matrix pencil may not have a complete set of eigenvectors. As an example, consider the pencil

which has one eigenvalue (multiplicity 2) and only one eigenvector, .

In §2.3, we know that when is positive definite we can find a matrix of eigenvectors and a diagonal matrix of eigenvalues such that with . The inner product forms a true inner product and is a norm.

When is indefinite and nonsingular and if is not defective (i.e., no eigenvectors are missing), we can find a full set of eigenvectors, . The equation still holds, and the eigenvectors can be chosen so that , where is a diagonal matrix with and on the diagonal (note that it is transpose, not conjugate transpose, even though some vectors can be complex). The inner product is an indefinite inner product or pseudo-inner product, and can be used for normalizing purposes. Unlike the positive definite case, there is a set of vectors having pseudolength zero (as measured by ). In fact, it is possible for an eigenvector to satisfy . This implies that the Rayleigh quotient

is undefined. This phenomenon is illustrated by the diagonal matrices

The vector is an eigenvector of corresponding to the multiple eigenvalue , but .

The analogy between the definite and the indefinite cases can be taken further. When is positive definite, is an approximate eigenvector, and is an approximate eigenvalue, we have the standard residual bound:

where is the norm with respect to , i.e., (similarly for the other norm). Also see §5.7. When is indefinite and nonsingular, , and is not defective, one can prove that

note that in the definite case. A similar bound is given by

Even if these bounds are not computable, they imply that a small residual is good. When is singular there are similar bounds; see [161].

If both and are singular, or close to singular, worse problems may occur. Assume that there is a nonzero vector such that ; then any complex number is an eigenvalue. A more general case is illustrated in the example below:

The characteristic polynomial is identically equal to zero, so any complex number is an eigenvalue; we have a singular matrix pencil. The null spaces of and are one-dimensional and are spanned by and , respectively. Note that intersection between the null spaces is just the zero vector. The singularity of the problem implies that (and ). To solve a singular problem using numerical methods the singularity must be removed. It is a hopeless task to attack the original problem, as can be seen by perturbing and . Set and add to , then . So if or is nonzero this is no longer a singular pencil. Taking then for any we have the eigenvalues , and . If and the eigenvalues are , and . When the eigenvalues are roughly and . Note that the perturbations can be made arbitrarily small. A general discussion about singular pencils can be found in §8.7.

In practice, if we have a singular pencil, is singular for any , and a good LU factorization routine should give a warning (when used in (a) and (b) in the following §8.6.2). Another sign of a singular pencil is that the eigenvalue routine produces some random eigenvalues at each run on the same problem.

Next: Algorithm Up: Symmetric Indefinite Lanczos Method Previous: Symmetric Indefinite Lanczos Method   Contents   Index
Susan Blackford 2000-11-20