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!