Without loss of gernerality, we can only consider $n$-dimensional diagonal matrix $M$ whose elements are all nonnegative, i.e. $$M=\operatorname{diag}(m_1,m_2,\cdots,m_n)\ (m_i \geq 0).$$ Then is there any $U,V\in U(n)$ such that $$UMV=\sum_{k=0}^{n-1}c_kS^k,$$ where $$S=\left( \begin{array}{cccc} 0 & 1 & & \\ & 0 & \ddots & \\ & & \ddots & 1 \\ & & & 0 \\ \end{array} \right),\ c_i \in \mathbb{C}$$ It is true for $n=2$, but what about a general $n$?

Note: This problem is arised in quantum theory, where unitary transform can be ignored. So, we want to find some simple representation and we do not know whether the above question is true or false. Any help or suggestion will be appreciated!

prescribedsingular values". It is clear that it is all that is needed (see any linear algebra textbook). But why are you so sure it is always possible? – fedja Aug 17 '13 at 17:17