Introduction

In theory and practical applications one may encounter eigenproblems that are more complicated than the standard and generalized eigenproblems discussed in previous chapters. In §9.2 we pay attention to the important class of quadratic eigenproblems, with a small sidestep to higher order polynomial eigenproblems in §9.3.

In §9.4 the scope of the algebraic eigenvalue problem is widened to geometrical properties of invariant subspaces. A template is presented that can be used to solve variational problems that are defined over sets of subspaces. More specifically, algorithms and software are provided for the optimization of a real valued function $F(Y)$ over $Y$ such that $Y^\ast Y=I$. It is shown, by example, how many eigenvalue-related problems, such as the Procrustes problem, determination of nearest Jordan structure, and trace minimization, can be attacked with the given techniques.