Given diagonal $n \times n$ matrix $D$, find $n \times m$ matrix $U$ and $m \times n$ matrix $V$ such that $$U V = D$$
I am trying to apply the Woodbury matrix identity. I am inverting $(A+UV)$, where the update to the matrix $A$ is a diagonal matrix. Thus, $UV$ is a diagonal matrix. I would like to be able to define the matrix $U$ and $V$ such that I can place elements at any position along the diagonal of the product $UV$. In other words, if I want to place a certain value, say a, on the diagonal say at row $3$ column $3$, what are the matrices $U$ and $V$. The diagonal of the product of $UV$ will have many zeros and a few particular non-zero values placed at desired locations along the diagonal. For example, what is $U$ and $V$ to generate a diagonal matrix with the following diagonal ($0,0,0,1,1,1,0,0,0$), or ($1,1,1,0,0,0,0,0,0$)? Thanks for your help!