## 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.