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Regular Versus Singular Problems

Let us start by considering the generalized eigenvalue problem , where

The eigenvalues of are the pairs , ) and , ) with the associated eigenvectors and , respectively. If is nonzero, then is a finite eigenvalue. Otherwise, if is zero, then is an eigenvalue of the matrix pair . But what happens if, for example, ? Then is zero for all , which means that we have a singular eigenvalue problem. In this case we have ; i.e., and have a common null space. We say that is an eigenvector for an indeterminate eigenvalue . Note that the common null space is a sufficient but not necessary condition to have a singular eigenvalue problem.

The most common generalized eigenvalue problems are regular; i.e., and are square matrices and the characteristic polynomial is only vanishing for a finite number of values, where denotes the degree of the polynomial. The corresponding is called a regular matrix pencil. The eigenvalues of a regular pencil are points in the extended complex plane . The eigenvalues are defined as the zeros of and additional eigenvalues.

An alternative representation of a matrix pencil is the cross product form: the set of matrices where . The mapping shows the relation between the eigenvalues of and . For example, zero and infinite eigenvalues are represented as and , respectively, and can be treated as any other points in .

If (and ) is identically zero for all (and pairs (, )), then is called singular and is a singular matrix pair. Singularity of is signaled by some . In the presence of roundoff, and may be very small. In these situations, the eigenvalue problem is very ill-conditioned, and some of the other computed nonzero values of and may be indeterminate. Such problems are further discussed and illustrated by examples in §8.7.4. Moreover, rectangular matrix pairs are singular and the corresponding is a singular pencil.

Next: Kronecker Canonical Form Up: Singular Matrix Pencils   Previous: Singular Matrix Pencils     Contents   Index
Susan Blackford 2000-11-20