The generalised problem is as follows: Is there a condition on a symmetric positive-semi-definite matrix $A$ that ensures that it has a positive eigenvector, i.e. an eigenvector $Av = \lambda v$ such that $ v_i > 0$ for all $i$? Or, indeed, even a weakly positive eigenvector, i.e. an eigenvector $Av = \lambda v$ such that $ v_i \geq 0$ for all $i$? I am aware of the Perron-Frobenius theorem, but the assumptions for that theorem are too strong.
The problem I actually want to solve is more specific, in case the extra information helps. If $B$ is the incidence matrix of a directed graph $G$, then $B$ can be viewed as the linear transformation whose input is vertex weightings and whose output is the corresponding edge weightings obtained by the vertex weight differences. Then the matrix that I want to understand is $A := BVB^T$, for a diagonal matrix with positive entries $V$ (for comparison, the graph Laplacian is $\Delta = B^TB$). This is a map from edge weightings to edge weightings. When does this admit a positive eigenvector?
Thanks to everyone in advance.
Best Answer
For the general case, note the following simple principle. For a vector $x$, write $\text{supp}(x) = \{i \mid x_i \neq 0\}$.
In particular, if $A$ has a strictly positive eigenvector with eigenvalue $\lambda$, then $A$ does not have a non-negative eigenvector with eigenvalue $\mu \neq \lambda$. Conversely, if the $\lambda$-eigenspace of $A$ contains a non-negative eigenvector but not a strictly positive vector, then $A$ does not have a strictly positive eigenvector.
For the remainder of this answer, I will focus on your more specific question.
First we look at eigenvectors with eigenvalue $0$. Here we show that a non-negative eigenvector exists if and only if $G$ has a directed cycle, and a similar statement for strictly positive eigenvectors (see Proposition 3 below). For this we use the following lemma.
Second, we look at eigenvectors with non-zero eigenvalue. Here things are the other way around: we show that a strictly positive eigenvector can only exist if $G$ is acyclic (see Proposition 7 below). For this we use the following lemma.
Next, we make a slight change of perspective. Henceforth, we let $G$ be an undirected graph with Laplacian $\Delta$, and we will look at all possible orientations of the edges of $G$.
If the orientation $o$ is fixed, then it is unclear whether or not there exists a compatible eigenvector of $V\Delta$. However, note that $V\Delta$ only depends on the undirected graph, and not on the orientation. If we choose a fixed eigenvector of $V\Delta$, then it is easy to see which orientations are compatible.
The following heuristic argument suggests that $BVB^T$ does not have a non-negative eigenvector for “most” orientations of $G$. If $V\Delta$ is sufficiently generic (specifically, if it has has $n$ different eigenvalues and if each eigenvector has $n$ different entries), then $G$ has at most $2n$ different acyclic orientations which admit a compatible eigenvector of $V\Delta$. (An eigenvector $y$ with $n$ different entries admits exactly one compatible orientation, and $-y$ is compatible with the reverse orientation. Different eigenvectors might give rise to the same orientation.) The following result of Linial shows that the total number of acyclic orientations is much larger than $2n$:
This suggests that “most” directed acyclic graphs do not have a non-negative eigenvector of $BVB^T$. (This argument is heuristic in the sense that it relies on genericity of $V\Delta$. If some eigenspace of $V\Delta$ has dimension $\geq 2$, then there are infinitely many eigenvectors, so there might be many more acyclic orientations which are compatible with some eigenvector of $V\Delta$.)
References.
[Lin86]: N. Linial, Legal colorings of graphs, Combinatorica 6(1) (1986) 49–54.