Abstract
In this paper, the unique weighted polar decomposition theorem for rectangular complex matrices is first proved based on introducing the weighted partial isometric matrices. Then the simultaneous weighted polar decomposition of two rectangular complex matrices is studied, for which two sufficient conditions and one criterion are proposed. In order to obtain further characteristics of the simultaneous weighted polar decomposition, a new partial ordering called WGL partial ordering is defined on the set of rectangular matrices. Some basic properties of this new partial ordering are derived. In addition, we also provide methods for computing the weighted polar decomposition and discuss error bounds for the approximate generalized positive semidefinite polar factor.
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