I am looking for a property on a polar decomposition of a specific kind of matrix.

Let $A$ be an $n \times n$, non-symmetric, real, weak diagonally dominant M-Matrix. Its diagonal is strictly positive, its off-diagonal is negative or zero and all its columns sum to zero. $A$ has rank $n-1$

$A$ has a polar decomposition $A=UP$ and a singular value decomposition $A = W \Sigma V^T$ where $U=WV^T$, $P=V \Sigma V^T$ and $U$, $V$ and $W$ are unitary matrices (i.e. $UU^T=VV^T=WW^T=I$).

Because $A$ has rank $n-1$, $V $ and $W $ are not unique.

Can I always find $V$ and $W$ such that $U$ is positive semi-definite (i.e. its eigenvalues have positive real part)?