# equivalent?

• Jun 1st 2010, 10:40 AM
Mauritzvdworm
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
• Jun 2nd 2010, 12:32 AM
Swlabr
Quote:

Originally Posted by Mauritzvdworm
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?)
• Jun 2nd 2010, 02:08 AM
Mauritzvdworm
* 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?
• Jun 2nd 2010, 02:37 AM
Opalg
Quote:

Originally Posted by Mauritzvdworm
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.