H-selfadjoint roots of H-selfadjoint matrices and H-polar decompositions over the quaternions

Abstract

All vector spaces in this thesis will be endowed with an indefinite inner product defined by an invertible Hermitian matrix H. We study cases where the H has either complex or quaternion entries depending on the context. We generalize known results on the existence of an H-selfadjoint square root of an H-selfadjoint complex matrix to the existence of an H-selfadjoint mth root. It is found that the conditions on an H-selfadjoint complex matrix B ensuring the existence of an H-selfadjoint mth root, are only necessary for the blocks with eigenvalue zero and the blocks with negative eigenvalues. The results are given as conditions on the canonical form of (B,H) and although the results associated with the negative eigenvalues are the same as in the square root case, the results associated with the zero eigenvalue are somewhat more intricate. A construction for an H-selfadjoint mth root is included in each of the proofs for different cases depending on the eigenvalues. The study of square roots of H-nonnegative complex matrices is interesting because of the simple structure of these matrices. For each of the three cases, namely for square roots, H-selfadjoint square roots and H-nonnegative square roots of H-nonnegative matrices, we give necessary and sufficient conditions for the existence of a square root and we describe the square roots for the blocks with zero eigenvalue. The Jordan normal form is obtained for a square root of an H-nonnegative matrix and for the cases where A is an H-selfadjoint square root or an H-nonnegative square root of an H-nonnegative matrix, a canonical form of (A,H) is obtained. Conditions are also found for an H-nonnegative square root to be stable. We extend the complex case of H-selfadjoint mth roots to the skew field of quaternions and using the complex matrix representation of quaternion matrices, we prove that the results are essentially the same in the quaternion case as in the complex case, despite the noncommutativity of quaternions. Due to a logical connection between square roots and polar decompositions, it is natural to study H-polar decompositions of quaternion matrices. We show that a quaternion matrix X admits an H-polar decomposition, say X = UA for an H-selfadjoint matrix A and an H-unitary matrix U, if and only if the matrix X[∗]X has an H-selfadjoint square root A and the null spaces of X and A coincide. Specialising the conditions we found for the existence of an H-selfadjoint mth root to the case m = 2, we conclude by giving the conditions in terms of the canonical form of the pair (X[∗]X,H) and a basis for the null space of X. We also prove that Witt’s theorem is true for quaternion matrices.