Given a matrix ${\bf{H}}\in \mathbb{C}^{M\times N}$, and suppose its singular value decomposition as ${\bf{H}}={\bf{U}}{\bf{\Sigma}}{\bf{V}}^H$, where ${\bf{U}}\in \mathbb{C}^{M\times M}$ and ${\bf{V}}\in \mathbb{C}^{N\times N}$ are two unitatary matrices and ${\bf{\Sigma}}$ is an $M\times N$ rectangular diagonal matrix with non-negative real numbers on the diagonal.
My question is: does there exist a specifical decompostion pair ${\bf{U}}$ and ${\bf{V}}$, satisfying ${\bf{U}}$ is an identity matrix, i.e., ${\bf{U}}={\bf I}$? If it exists, then how to find the corresponding ${\bf{V}}$?