Matrix eigenvalue problems arise from practical applications usually only

after a long process of simplifications, discretizations and

linearizations. In most cases, the resulting matrices are highly

structured. For example, matrix representations may contain redundancy,

often in the form of sparsity, or inherit some physical properties from

the original problem. Aimed at a wider audience, the purpose of this talk

is to summarize some of the existing knowledge on the theoretical and

computational treatment of such structured eigenvalue problems. Particular

attention will be paid to the notion of structured condition numbers,

which provide a first-order measure on the sensitivity of an eigenvalue or

invariant subspace under perturbations that respect the matrix structure.

A general framework covering Lie groups, Lie algebras and Jordan algebras

associated with bilinear and sesquilinear forms is presented.

This is joint work with Peter Benner, Ralph Byers, Heike Fassbender,

Michael Karow and Francoise Tisseur.