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 less than $n$.

$A$ has a singular value decomposition $A = W \Sigma V^T$ where $W$ and $V^T$ are unitary matrices (i.e. $VV^T=WW^T=I$).

Because $A$ is low rank, $V^T$ and $W$ are not unique.

Is the following assessment true?:

For any $n \times n$, non-symmetric, real, weak diagonally dominant, low rank M-Matrix $A$ there exists a singular value decomposition $A = W \Sigma V^T$ such that $WV^T$ is positive semi-definite (i.e. its eigenvalues have positive real part).

Example:

$A = \left( \begin{array}{ccc} 1 & -1 & -2 \\ 0 & 2 & -1 \\ -1 & -1 & 3 \end{array} \right), W = \left(\begin{array}{ccc} -0.46 & 0.67 & 0.58 \\ -0.35 & -0.74 & 0.58 \\ 0.81 & 0.064 & 0.58 \end{array} \right),$ $ V^T=\left(\begin{array}{ccc} -0.31 & -0.26 & 0.91 \\ 0.26 & -0.94 & -0.18 \\ 0.91 & 0.18 & 0.37 \end{array} \right), \Sigma=\left(\begin{array}{ccc} 4.1 & 0 & 0 \\ 0 & 2.3 & 0 \\ 0 & 0 & 0 \end{array} \right)$

Eigenvalues of $WV^T$: $0.84+0.54i, 0.84-0.54j, 1$

Changing the sign of the third row of $V^T$ conserves the properties of all matrices but the last eigenvalue of $WV^T$ changes from $1$ to $-1$ making $WV^T$ lose positive definiteness.

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 less than $n$.

$A$ has a singular value decomposition $A = W \Sigma V^T$ where $W$ and $V^T$ are unitary matrices (i.e. $VV^T=WW^T=I$).

Because $A$ is low rank, $V^T$ and $W$ are not unique.

Is the following assessment true?:

For any $n \times n$, non-symmetric, real, weak diagonally dominant, low rank M-Matrix $A$ there exists a singular value decomposition $A = W \Sigma V^T$ such that $WV^T$ is positive semi-definite (i.e. its eigenvalues have positive real part).

Example:

$A = \left( \begin{array}{ccc} 1 & -1 & -2 \\ 0 & 2 & -1 \\ -1 & -1 & 3 \end{array} \right), W = \left(\begin{array}{ccc} -0.46 & 0.67 & 0.58 \\ -0.35 & -0.74 & 0.58 \\ 0.81 & 0.064 & 0.58 \end{array} \right),$ $ V^T=\left(\begin{array}{ccc} -0.31 & -0.26 & 0.91 \\ 0.26 & -0.94 & -0.18 \\ 0.91 & 0.18 & 0.37 \end{array} \right), \Sigma=\left(\begin{array}{ccc} 4.1 & 0 & 0 \\ 0 & 2.3 & 0 \\ 0 & 0 & 0 \end{array} \right)$

Eigenvalues of $WV^T$: $0.84+0.54i, 0.84-0.54j, 1$

Changing the sign of the third row of $V^T$ conserves the properties of all matrices but the last eigenvalue of $WV^T$ changes from $1$ to $-1$ making $WV^T$ lose positive definiteness.

EDIT 1

The following papers seem to address part of the problem:

A. Horn, R. Steinberg. Eigenvalues of the unitary part of a matrix. Pacific Journal of Mathematics Vol. 9, No. 2, June, 1959, pp. 541-550

A. Horn, On the eigenvalues of a matrix with prescribed singular values, Proc. Amer. Math. Soc, 5 (1954), 4-7.

R. Horn, G. Piazza, T. Politi. Explicit polar decompositions of complex matrices. Electronic Journal of Linear Algebra. Vol. 18, pp. 693-699, November 2009

The matrix $W V^T$ is also called phase gain in the context of control systems.

The answer to the problem seems to be there but eludes me still...

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)?

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 less than $n$.

$A$ has a singular value decomposition $A = W \Sigma V^T$ where $W$ and $V^T$ are unitary matrices (i.e. $VV^T=WW^T=I$).

Because $A$ is low rank, $V^T$ and $W$ are not unique.

Is the following assessment true?:

For any $n \times n$, non-symmetric, real, weak diagonally dominant, low rank M-Matrix $A$ there exists a singular value decomposition $A = W \Sigma V^T$ such that $WV^T$ is positive semi-definite (i.e. its eigenvalues have positive real part).

Example:

$A = \left( \begin{array}{ccc} 1 & -1 & -2 \\ 0 & 2 & -1 \\ -1 & -1 & 3 \end{array} \right), W = \left(\begin{array}{ccc} -0.46 & 0.67 & 0.58 \\ -0.35 & -0.74 & 0.58 \\ 0.81 & 0.064 & 0.58 \end{array} \right),$ $ V^T=\left(\begin{array}{ccc} -0.31 & -0.26 & 0.91 \\ 0.26 & -0.94 & -0.18 \\ 0.91 & 0.18 & 0.37 \end{array} \right), \Sigma=\left(\begin{array}{ccc} 4.1 & 0 & 0 \\ 0 & 2.3 & 0 \\ 0 & 0 & 0 \end{array} \right)$

Eigenvalues of $WV^T$: $0.84+0.54i, 0.84-0.54j, 1$

Changing the sign of the third row of $V^T$ conserves the properties of all matrices but the last eigenvalue of $WV^T$ changes from $1$ to $-1$ making $WV^T$ lose positive definiteness.