Suppose $U$ is a unitary matrix. Does there exist a non-singular matrix $R$ such that $$ U=\bar{R}R^{-1}? $$ The bar denotes the complex conjugation of each entry of the matrix $R$.
Remarks: 1) It is obviously true for $1\times 1$. 2) If $U$ admits the decomposition $U=VDV^{T}$, where $V$ is orthonormal and $D$ diagonal with diagonal entries of unit modulus, then $R:=VD^{-1/2}V^{T}$.