Call a (not necessarily square) nonnegative matrix $M$ connected if there do not exist permutation matrices $P$ and $Q$ such that $PMQ=\begin{pmatrix}A&0\\0&B\end{pmatrix}$ for some $A$ and $B$. [Perhaps this property already has a standard name.]

I'm looking for a proof of the following (assuming it's true, as I believe):

If a matrix is (nonnegative and) connected, then

1. its dominant (greatest) singular value $s$ is not degenerate (i.e. it has unique left and right singular vectors),
2. the left and right singular vectors of $s$ are positive (not just nonnegative).

My guess is that this is probably relatively straightforward, or even follows directly from some standard results. However, I'm a combinatorialist with limited linear algebra, so would appreciate some help.

Call a (not necessarily square) nonnegative matrix $M$ connected if there do not exist permutation matrices $P$ and $Q$ such that $PMQ=\begin{pmatrix}A&0\\0&B\end{pmatrix}$ for some $A$ and $B$. [Perhaps this property already has a standard name.]

I'm looking for a proof of the following (assuming it's true, as I believe):

If a matrix is (nonnegative and) connected, then

1. its dominant (greatest) singular value $s$ is not degenerate (i.e. it has unique left and right singular vectors),
2. the left and right singular vectors of $s$ are positive (not just nonnegative).

My guess is that this is probably relatively straightforward, or even follows directly from some standard results. However, I'm a combinatorialist with limited linear algebra, so would appreciate some help.

The context is this paper by Albert & Vatter.