Structures matrices in indefinite inner product spaces : simple forms, invariant subspaces and rank–one perturbations

Thesis (PhD (Mathematics))--North-West University, Potchefstroom Campus, 2012

The (definite) inner product between P two vectors x; y 2 Rn is defined by (x,y) = [not able to show]. The length of a vector x 2 Rn is then described by the inner product as [not able to show]. In this thesis the definite inner product is replaced by an indefinite inner product. This has a substantial impact on the geometry of sub-spaces. The indefinite inner product between two vectors in Rn can uniquely be represented by [x; y] = hHx; yi, for a real invertible symmetric matrix H = HT. In the literature there exists an extensive theory for the classes of hermitian, unitary and normal matrices for the definite inner product space. Similar classes are also studied in the context of indefinite inner product space. In this thesis we focus on a study of H-positive real and H-expansive matrices. This study is closely related to a thorough study of H-dissipative and H-contractive matrices, which already exists in the literature. In Chapter 2 we discuss a class of matrices that is closely related to H-dissipative matrices, namely, the class of H-positive real matrices. A matrix A is H-dissipative if [not able to show]. A non-complex matrix A is H-positive real if [not able to show]. It is easily seen that A is H-positive real if and only if iA is H-dissipative, and hence it follows that -iA is H-positive real if and only if A is H-dissipative. A simple form for the matrix H is obtained and thereafter the A-invariant maximal H-nonnegative and H-non-positive sub-spaces are constructed. The main focus however, is to obtain the uniqueness and stability of these sub-spaces. The numerical range condition is used for this purpose. Both the real and complex cases are treated in this chapter. In Chapter 3 a second class of matrices in the indefinite inner product space is considered, namely the class of H-expansive matrices. These are matrices A for which A_HA*H is nonnegative. As in the previous chapter, our purpose is the construction of complex (as well as real) A-invariant maximal H-nonnegative and A-invariant maximal H-non-positive sub-spaces. Alternatively, we could have used a suitable Cayley transform to obtain the same results, except for the case when A is real and has both the eigenvalues 1 and -1. The techniques developed in this thesis covers this case. The techniques again entail that a simple form for the matrix pair (A,H) is obtained, with A in Jordan canonical form or in the real canonical form. The simple form is then used to construct the A-invariant maximal H-nonnegative and A-invariant maximal H-non-positive sub-spaces. The last section of the chapter is devoted to obtaining a simple form for the class of H-unitary matrices. We say a real matrix A is H-unitary if ATHA-H = 0, for any real symmetric invertible matrix H. What complicates matters here are if A has eigenvalues +-1 and the corresponding Jordan blocks are of even size. In the final chapter of the thesis a different topic is considered, namely, rank-one perturbations of H-positive real matrices. A rank-one perturbation of a matrix A is given by B = A + uv*, where u and v are in Cn. We mainly consider perturbations that have the additional structure that uv* is itself H-positive real. We are interested in the generic behaviour of eigenvalues under such structured rank-one perturbations. A new and interesting result is obtained; that eigenvalues introduced by the perturbation cannot lie on the imaginary axis. However, asymptotically, they may tend towards the imaginary axis. The chapter ends with a few examples which illustrate an interesting phenomenon which occurs when considering rank-one perturbations in the real case.

Doctoral