Polar decompositions of quaternion matrices in indefinite inner product spaces

TL;DR

This paper investigates the conditions for polar decompositions of quaternion matrices within indefinite inner product spaces, extending classical results and providing new criteria for their existence.

Contribution

It establishes necessary and sufficient conditions for $H$-polar decompositions of quaternion matrices and extends Witt's theorem to quaternion matrices in this context.

Findings

Derived conditions for $H$-polar decomposition existence

Extended Witt's theorem to quaternion matrices

Provided theoretical framework for indefinite inner product spaces

Abstract

Polar decompositions of quaternion matrices with respect to a given indefinite inner product are studied. Necessary and sufficient conditions for the existence of an $H$-polar decomposition are found. In the process an equivalent to Witt's theorem on extending $H$-isometries to $H$-unitary matrices is given for quaternion matrices.