Abstract
A number of necessary and sufficient conditions are given for the existence of unitary matrices U and V, such that UAV is a diagonal matrix for every matrix A in some set Γ of rectangular complex matrices. Two related questions are then considered. A necessary and sufficient condition for the existence of unitary matrices U and V such that UAV is a real diagonal matrix for every A in Γ is obtained, and an improvement on a necessary and sufficient condition discovered by R.C. Thompson for the existence of real orthogonal matrices P and Q such that PAQ is a diagonal matrix for every A in Γ is given.
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