Madelein van Straaten is defending her joint PhD between North-West University in South Africa and the Vrije Universiteit under the supervision of prof. dr. G.J. Groenewald and prof. dr. A.C.M. Ran.
The central topic of her research is the study of matrices in indefinite inner product spaces. Let H be the invertible Hermitian matrix defining an indefinite inner product. If a given matrix B is H-selfadjoint, necessary and sufficient conditions are found such that there exists an H-selfadjoint m-th root A of the matrix B, that is, Am=B. This is studied for matrices with complex entries and for matrices with quaternion entries. A construction for an H-selfadjoint m-th root is included per eigenvalue case. Square roots of H-nonnegative matrices are also studied together with their stability. The last part of this research is on polar decompositions of quaternion matrices in indefinite inner product spaces and it includes a proof of Witt’s theorem for quaternion matrices.
The thesis can be found here.