18. Finite-Dimensional Indefinite Inner Product Spaces and Applications in Numerical Analysis

The aim of this chapter is to give a few examples for the fruitful interaction of the theory of finite-dimensional indefinite inner product spaces as a special theme in Operator Theory on the one hand and Numerical Linear Algebra as a special theme in Numerical Analysis on the other hand. Two particular topics are studied in detail. First, the theory of polar decompositions in indefinite inner product spaces is reviewed, and the connection between polar decompositions and normal matrices is highlighted. It is further shown that the adaption of existing algorithms from Numerical Linear Algebra allows the numerical computation of these polar decompositions. Second, two particular applications are presented that lead to the Hamiltonian eigenvalue problem. The first example deals with Algebraic Riccati Equations that can be solved via the numerical computation of the Hamiltonian Schur form of a corresponding Hamiltonian matrix. It is shown that the question of the existence of the Hamiltonian Schur form can only be completely answered with the help of a particular invariant discussed in the theory of indefinite inner products: the sign characteristic. The topic of the second example is the stability of gyroscopic systems, and it is again the sign characteristic that allows the complete understanding of the different effects that occur if the system is subject to either general or structure-preserving perturbations.