1. ## 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

2. 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?)

3. * 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?

4. 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.