Asked this on MathOverflow with some more details
I have a matrix originating from Master Equation for birth death process on semi infinite lattice. It is tridiagonal. It is not symmetric.
I wanted to know if there is any way to compute the elements of $e^{(H-\lambda_1 I)t}$ where $\lambda_1$ is the first eigenvalue of $H$ and $I$ is the identity matrix. (analytically. I know perhaps I could try to numerically approximate but I am hoping to use that as last resort)
Any help is appreciated! I can provide more details if needed.
Edit: Additional details
The matrix I have looks like this: \begin{pmatrix} 0&d&0&0&.&.&.\\ 0&-(b+d)&2d&0&.&.&.\\ 0&b&-(2b+2d)&3d&0&.&.\\0&0&2b&-(3b+3d)&4d&0&.\\ .&.&0&3b&-(4b+4d)&5d&.\\.&.&.\end{pmatrix}
$b$ and $d$ are birth and death rates and both are positive. This is an infinite matrix since it is over all possible lattice sites. I don't know of any approximate methods either. That would be helpful too. This is a sparse tridiagonal matrix. But I don't know if I can apply the approximate methods that exist for finite matrices.
PS: From writing the generating function for the probability distribution and calculating it at long times, I know the first eigenvalue and the first eigenstate. But it is too complicated to write an expansion of the generating function for all times. Hence, I do not know how the other eigenstates look like. I also know that the zeroth eigenvalue is 0 and the corresponding eigenstate. Maybe that would help somehow?