SVD decomposition to UV decomposition

Question

Let's say I have an SVD decomposition of a matrix $M$ of size $m\times n$: $M = A\sum B$ where $A:m\times m, \sum:m\times n, B: n\times n$. I want to convert it to $M = UV$ where $U:m\times d$, $V: d\times n$. What are the algorithms to do so ? Are there any parallel ones ?

I have one idea is that first, UV decompose the diagonal matrix $\sum$ into two matrices $\sum = L\times L'$ where $L:m\times d, L'=d\times n$. Then, $U = AL$ and $V=L'B$. But this requires UV decomposition of a diagonal matrix.

Edit

d is a predefined constant and $d < min(m,n)$.

Thanks.

Answer 1

You can use Cholesky Factorization instead, if you don't want to use $UV$ decomposition. But seems you still need to know more about the SVD itself. I think this may help: http://www.cs.iastate.edu/~cs577/handouts/svd.pdf

Answer 2

Yes, it's good to check the conditioning first. Here is an easy way to check ill-conditioning: Check the condition number of your matrix using $cond(A)=||A||||A^{-1}||$. When this number is very large then your matrix is ill-conditioned and you can use any norm for that.