sum of singular vector dyadics derived from the matrix itself

Question

I have an m*n (m>n) n-rank matrix (let's denote it by A), with nonnegative elements. SVD decomposition says, that A=UDV', where U and V are orthogonal matrixes, and their columns are the singular vectors. I would like to express UV' by only using matrix A and the singular values, but my algebra knowledge not enough. Do you have any idea?

Answer 1

Assuming that $U$ is $m\times n$ and $V$, $D$ are $n\times n$:

You have $A = UDV^*$, thus $A^TA = VD^2V^*$. Hence $(A^TA)^{-1/2} = VD^{-1}V^*$. Hence, $A(A^TA)^{-1/2} = AVD^{-1}V^* = UDD^{-1}V^* = UV^*$. So, you have $UV^*$ expressed only in terms of $A$. Maybe this helps.