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Thread: equivalent?

  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 $\displaystyle \mathbb{I}_{n}=U^*U$ where * denotes transpose complex conjugation and that $\displaystyle \mathbb{I}_m=UU^*$ where $\displaystyle \mathbb{I}_m$ 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 $\displaystyle \mathbb{I}_{n}=U^*U$ where * denotes transpose complex conjugation and that $\displaystyle \mathbb{I}_m=UU^*$ where $\displaystyle \mathbb{I}_m$ 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 $\displaystyle m\neq n$ 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 $\displaystyle \mathbb{I}_{n}=U^*U$ where * denotes transpose complex conjugation and that $\displaystyle \mathbb{I}_m=UU^*$ where $\displaystyle \mathbb{I}_m$ denotes the mxm identity matrix
    Not unless m=m, because U*U and UU* have the same rank, but $\displaystyle \mathbb{I}_{n}$ has rank n and $\displaystyle \mathbb{I}_{m}$ has rank m.
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