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  1. #1
    Member Mauritzvdworm's Avatar
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    equivalent?

    Is it possible to find a rectangular matrix U such that we can write the nxn identity matrix as \mathbb{I}_{n}=U^*U where * denotes transpose complex conjugation and that \mathbb{I}_m=UU^* where \mathbb{I}_m denotes the mxm identity matrix
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  2. #2
    MHF Contributor Swlabr's Avatar
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    Quote Originally Posted by Mauritzvdworm View Post
    Is it possible to find a rectangular matrix U such that we can write the nxn identity matrix as \mathbb{I}_{n}=U^*U where * denotes transpose complex conjugation and that \mathbb{I}_m=UU^* where \mathbb{I}_m denotes the mxm identity matrix
    What is the context? Can you not just set m=n=1...(although...what do you mean by complex conjugation w.r.t. matrices?)
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  3. #3
    Member Mauritzvdworm's Avatar
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    * is taking the transpose of the matrix and taking the complex conjugate of the entries of the matrix, for the case where n=m, it is trivial, but what about the m\neq n case?
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  4. #4
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    Quote Originally Posted by Mauritzvdworm View Post
    Is it possible to find a rectangular matrix U such that we can write the nxn identity matrix as \mathbb{I}_{n}=U^*U where * denotes transpose complex conjugation and that \mathbb{I}_m=UU^* where \mathbb{I}_m denotes the mxm identity matrix
    Not unless m=m, because U*U and UU* have the same rank, but \mathbb{I}_{n} has rank n and \mathbb{I}_{m} has rank m.
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