equivalent?

**Mauritzvdworm** · June 1st 2010, 10:40 AM

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

**Swlabr** · June 2nd 2010, 12:32 AM

Originally Posted by Mauritzvdworm

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?)

**Mauritzvdworm** · June 2nd 2010, 02:08 AM

* 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?

**Opalg** · June 2nd 2010, 02:37 AM

Originally Posted by Mauritzvdworm

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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