Let $C$ be a $2 \times 2$, negative definite asymmetric matrix with real entries. The matrix $C$ has negative coefficients, except for a single positive one in a corner. Furthermore, assume that $d > 0$.
Define: $$ H(t) = e^{C t}\cdot \left[\begin{matrix}d \\ 0\end{matrix}\right]\in\mathbb{R}^2 $$ Denote $\min(H(t))<0$ if at least one of the elements is less than $0$.
Fixing $d > 0$, under what conditions (beyond negative definiteness) on $C$ and $d$ does a $t^{*}> 0$ exist such that $\min(H(t^*)) < 0$?
Note: As I need to prove that a negative $H(t)$ always exists in my setup, I would be perfectly happy if you were also able to show that this is not possible given the assumptions.
(this is related to When is there a vector $D$ with positive coordinates such that $e^{Ct}D$ has a negative coordinate? but with different assumptions, as we no longer use Perron and Frobenius since the matrix is not purely negative).