I have the following matrix equation:
$\mathbf{x}^\top e^{t\mathbf{A}}\mathbf{y}=c$
$\mathbf{x}$ and $\mathbf{y}$ are vectors of length $k$, $\mathbf{A}$ is a $k\times k$ matrix, and $c$ and $t$ are scalars. $e^{t\mathbf{A}}$ is the matrix exponential of $t\mathbf{A}$.
I want to solve for $t$. Is this possible analytically?
I don't think these extra details simplify the solution, but here they are anyway: $\mathbf{x}$ is a vector of probabilities that sum to $1$. $\mathbf{y}$ is a vector of zeroes except for the $i$th element which is $1$. $\mathbf{A}$ is a transition intensity matrix, whose rows sum to zero.