Decomposing a matrix with unit sphere constraints

Question

I would like to decompose an $m\times n$ matrix $A$ into two matrices $U\in\mathbb{R}^{m\times n}$ and $V\in\mathbb{R}^{n\times n}$ such that $UV=A$, and the $m$ rows of $U$ each have unit magnitude. In other words, I want the rows of $U$ to lie on the unit (hyper-)sphere. Is such a decomposition possible in general? Is there a formula or method for finding it?

Answer 1

Take any $X\in\mathbb{R}^{m\times n}$ with $\operatorname{rank}(X)=m\,$ and construct the semi-orthogonal matrix $$ U=(XX^T)^{-1/2}\,X \quad\implies\quad UU^T = I = UU^{\bf+} $$ Note that the pseudoinverse of $U$ is equal to its transpose.

Construct the second factor as $\; V = U^TA$

Then $\:UV=UU^TA=IA=A.\;$ This factorization is not unique, since the choice of $X$ was arbitrary.